GENERAL TOPOLOGY

JESPER M. MØLLER

Contents

1. Sets, functions and relations  3
1.1. Sets  3
1.2. Functions  3
1.5. Relations  4
2. The integers and the real numbers  7
3. Products and coproducts  9
4. Finite and infinite sets  10
5. Countable and uncountable sets  12
6. Well-ordered sets  14
7. Partially ordered sets, The Maximum Principle and Zorn’s lemma  16
8. Topological spaces  18
8.4. Subbasis and basis for a topology  18
9. Order topologies  20
10. The product topology  20
10.4. Products of linearly ordered spaces  21
10.6. The coproduct topology  21
11. The subspace topology  22
11.5. Subspaces of linearly ordered spaces  22
12. Closed sets and limit points  24
12.3. Closure and interior  24
12.10. Limit points and isolated points  25
12.14. Convergence, the Hausdorff property, and the T_{1}-axiom  26
13. Continuous functions  27
13.6. Homeomorphisms and embeddings  28
13.13. Maps into products  29
13.17. Maps out of coproducts  30
14. The quotient topology  31
14.1. Open and closed maps  31
14.2. Quotient topologies and quotient maps  31
15. Metric topologies  38
15.6. The first countability axiom  38
15.13. The uniform metric  40
16. Connected spaces  41
17. Connected subsets of linearly ordered spaces  43
17.5. Path connected spaces  44
17.9. Components and path components  44
17.12. Locally connected and locally path connected spaces  45
18. Compact spaces  48
19. Compact subspaces of linearly ordered spaces  51
19.6. Compactness in metric spaces  53
20. Limit point compactness and sequential compactness  54
21. Locally compact spaces and the Alexandroff compactification  54
22. Countability Axioms  58

Date: January 18, 2006.

23. Separation Axioms  59
24. Normal spaces  61
25. Second countable regular spaces and the Urysohn metrization theorem  63
25.1. An embedding theorem  63
25.5. A universal second countable regular space  64
26. Completely regular spaces and the Stone–Čech compactification  66
26.6. The Stone–Čech construction  66
27. Manifolds  69
28. Relations between topological spaces  70
References  71

1. Sets, functions and relations

1.1. Sets. A set is a collection of mathematical objects. We write a ∈ S if the set S contains the object a.

Example 1.1. The natural numbers 1, 2, 3, … can be collected to form the set Z_{+} = {1, 2, 3, …}.

This naïve form of set theory unfortunately leads to paradoxes. Russel’s paradox^{1} concerns the formula S ∉ S. First note that it may well happen that a set is a member of itself. The set of all infinite sets is an example. The Russel set

R = {S | S ∉ S}

is the set of all sets that are not a member of itself. Is R ∈ R or is R ∉ R?

How can we remove this contradiction?

Definition 1.2. The universe of mathematical objects is stratified. Level 0 of the universe consists of (possibly) some atomic objects. Level i > 0 consists of collections of objects from lower levels. A set is a mathematical object that is not atomic.

No object of the universe can satisfy S ∈ S for atoms do not have elements and a set and an element from that set can not be in the same level. Thus R consists of everything in the universe. Since the elements of R occupy all levels of the universe there is no level left for R to be in. Therefore R is outside the universe, R is not a set. The contradiction has evaporated!

Axiomatic set theory is an attempt to make this precise formulating a theory based on axioms, the ZFC-axioms, for set theory. (Z stands for Zermelo, F for Fraenkel, and C for Axiom of Choice.) It is not possible to prove or disprove the statement ”ZFC is consistent” within ZFC – that is within mathematics [11].

If A and B are sets then

A ∩ B = {x | x ∈ A and x ∈ B}  A ∪ B = {x | x ∈ A or x ∈ B}
A × B = {(x, y) | x ∈ A and y ∈ B} A ⨿ B = {(1, a) | a ∈ A} ∪ {(2, b) | b ∈ B}

and

A − B = {x | x ∈ A and x ∉ B}

are also sets. These operations satisfy

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)  A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A − (B ∪ C) = (A − B) ∩ (A − C)  A − (B ∩ C) = (A − B) ∪ (A − C)

as well as several other rules.

We say that A is a subset of B, or B a superset of A, if all elements of A are elements of B. The sets A and B are equal if A and B have the same elements. In mathematical symbols,

A ⊂ B ⟺ ∀x ∈ A: x ∈ B
A = B ⟺ (∀x ∈ A: x ∈ B and ∀x ∈ B: x ∈ A) ⟺ A ⊂ B and B ⊂ A

The power set of A,

P(A) = {B | B ⊂ A}

is the set of all subsets of A.

1.2. Functions. Functions or maps are fundamental to all of mathematics. So what is a function?

Definition 1.3. A function from A to B is a subset f of A × B such that for all a in A there is exactly one b in B such that (a, b) ∈ f.

We write f: A → B for the function f ⊂ A × B and think of f as a rule that to any element a ∈ A associates a unique object f(a) ∈ B. The set A is the domain of f, the set B is the codomain of f; dom(f) = A, cod(f) = B.

The function f is • injective or one-to-one if distinct elements of A have distinct images in B,

• surjective or onto if all elements in B are images of elements in A, • bijective if is both injective and surjective, if any element of B is the image of precisely

1If a person says ”I am lying” – is he lying?

one element of A. In other words, the map f is injective, surjective, bijective iff the equation f(a) = b has at most one solution, at least one solution precisely one solution, for all b ∈ B.

If f: A → B and g: B → C are maps such that cod(f) = dom(g), then the composition is the map g ∘ f: A → C defined by g ∘ f(a) = g(f(a)).

Proposition 1.4. Let A and B be two sets.

(1) Let f: A → B be any map. Then

f is injective ⟺ f has a left inverse
f is surjective ⟺^{AC} f has a right inverse
f is bijective ⟺ f has an inverse

(2) There exists a surjective map A ↠ B ⟺^{AC} There exits an injective map B ↣ A

Any left inverse is surjective and any right inverse is injective.

If f: A → B is bijective then the inverse f^{−1} : B → A is the map that to b ∈ B associates the unique solution to the equation f(a) = b, ie

a = f^{−1}(b) ⟺ f(a) = b

for all a ∈ A, b ∈ B.

Let map(A, B) denote the set of all maps from A to B. Then

map(X, A × B) = map(X, A) × map(X, B), map(A ⨿ B, X) = map(A, X) × map(B, X)

for all sets X, A, and B. Some people like rewrite this as

map(X, A × B) = map(ΔX, (A, B)), map(A ⨿ B, X) = map((A, B), ΔX)

where ΔX = (X, X). Here, (A, B) is a pair of spaces and maps (f, g): (X, Y) → (A, B) between pairs of spaces are defined to be pairs of maps f: X → A, g: Y → B. These people say that the product is right adjoint to the diagonal and the coproduct is left adjoint to the diagonal.

1.5. Relations. There are many types of relations. We shall here concentrate on equivalence relations and order relations.

Definition 1.6. A relation R on the set A is a subset R ⊂ A × A.

Example 1.7. We may define a relation D on Z_{+} by aDb if a divides b. The relation D ⊂ Z_{+}×Z_{+} has the properties that aDa for all a and aDb and bDc ⟹ aDc for all a, b, c. We say that D is reflexive and transitive.

1.5.1. Equivalence relations. Equality is a typical equivalence relation. Here is the general definition.

Definition 1.9. An equivalence relation on a set A is a relation ∼⊂ A × A that is Reflexive: a ∼ a for all a ∈ A Symmetric: a ∼ b ⇒ b ∼ a for all a, b ∈ A Transitive: a ∼ b ∼ c ⇒ a ∼ c for all a, b, c ∈ A

The equivalence class containing a ∈ A is the subset

[a] = {b ∈ A | a ∼ b}

of all elements of A that are equivalent to a. There is a canonical map [ ]: A → A/∼ onto the set

A/∼= {[a] | a ∈ A} ⊂ P(A)

of equivalence classes that takes the element a ∈ A to the equivalence class [a] ∈ A/∼ containing a.

A map f: A → B is said to respect the equivalence relation ∼ if a_{1} ∼ a_{2} ⟹ f(a_{1}) = f(a_{2}) for all a_{1}, a_{2} ∈ A (f is constant on each equivalence class). The canonical map [ ]: A → A/∼ respects the equivalence relation and it is the universal example of such a map: Any map f: A → B that respects the equivalence relation factors uniquely through A/∼ in the sense that there is a unique map \overline{f} such that the diagram

\xymatrix{A \ar[rr]^{f} \ar[dr]_{[ ]} &  & B \\  & A/∼ \ar@{-->}[ur]_{∃!\overline{f}}}

commutes. How would you define \overline{f}?

Example 1.10. (1) Equality is an equivalence relation. The equivalence class [a] = {a} contains just one element.

(2) a mod b mod n is an equivalence relation on Z. The equivalence class [a] = a + nZ consists of all integers congruent to a mod n and the set of equivalence classes is Z/nZ = {[0], [1], …, [n−1]}.

(3) x ∼ y ⟺^{def} |x| = |y| is an equivalence relation in the plane R^{2}. The equivalence class [x] is a circle centered at the origin and R^{2}/ ∼ is the collection of all circles centered at the origin. The canonical map R^{2} → R^{2}/∼ takes a point to the circle on which it lies.

(4) If f: A → B is any function, a_{1} ∼ a_{2} ⟺^{def} f(a_{1}) = f(a_{2}) is an equivalence relation on A. The equivalence class [a] = f^{−1}(f(a)) ⊂ A is the fibre over f(a) ∈ B. we write A/f for the set of equivalence classes. The canonical map A → A/f takes a point to the fibre in which it lies. Any map f: A → B can be factored

[figure: A; f; [ ]; B; A/f; \overline{f}]

as the composition of a surjection followed by an injection. The corestriction \overline{f}: A/f → f(A) of \overline{f} is a bijection between the set of fibres A/f and the image f(A).

(5) [Ex 3.2] (Restriction) Let X be a set and A ⊂ X a subset. Declare any two elements of A to be equivalent and any element outside A to be equivalent only to itself. This is an equivalence relation. The equivalence classes are A and {x} for x ∈ X − A. One writes X/A for the set of equivalence classes.

(6) [Ex 3.5] (Equivalence relation generated by a relation) The intersection of any family of equivalence relations is an equivalence relation. The intersection of all equivalence relations containing a given relation R is called the equivalence relation generated by R.

Lemma 1.11. Let ∼ be an equivalence relation on a set A. Then (1) a ∈ [a]

(2) [a] = [b] ⟺ a ∼ b
(3) If [a] ∩ [b] ≠ ∅ then [a] = [b]

Proof. (1) is reflexivity, (2) is symmetry, (3) is transitivity: If c ∈ [a] ∩ [b], then a ∼ c ∼ b so a ∼ b and [a] = [b] by (2). □

This lemma implies that the collection A/∼ is a partition of A, a collection of nonempty, disjoint subsets of A whose union is all of A. Conversely, given any partition of A we define an equivalence relation by declaring a and b to be equivalent if they lie in the same subset of the partition. We conclude that an equivalence relation is essentially the same thing as a partition.

1.5.2. Linear Orders. The usual order relation < on Z or R is an example of a linear order. Here is the general definition.

Definition 1.12. A linear order on the set A is a relation <⊂ A × A that is Comparable: If a ≠ b then a < b or b < a for all a, b ∈ A Nonreflexive: a < a for no a ∈ A Transitive: a < b < c ⇒ a < c for all a, b, c ∈ A

What are the right maps between ordered sets?

Definition 1.13. Let (A, <) and (B, <) be linearly ordered sets. An order preserving map is a map f: A → B such that a_{1} < a_{2} ⟹ f(a_{1}) < f(a_{2}) for all a_{1}, a_{2} ∈ A. An order isomorphism is a bijective order preserving map.

An order preserving map f: A → B is always injective. If there exists an order isomorphism f: A → B, then we say that (A, <) and (B, <) have the same order type.

How can we make new ordered sets out of old ordered sets? Well, any subset of a linearly ordered set is a linearly ordered set in the obvious way using the restriction of the order relation. Also the product of two linearly ordered set is a linearly ordered set.

Definition 1.14. Let (A, <) and (B, <) be linearly ordered sets. The dictionary order on A × B is the linear order given by

(a_{1}, b_{1}) < (a_{2}, b_{2}) ⟺^{def} (a_{1} < a_{2}) or (a_{1} = a_{2} and b_{1} < b_{2})

The restriction of a dictionary order to a product subspace is the dictionary order of the restricted linear orders. (Hey, what did that sentence mean?)

What about orders on A ⨿ B, A ∪ B, map(A, B) or P(A)?

What are the invariant properties of ordered sets? In a linearly ordered set (A, <) it makes sense to define intervals such as

(a, b) = {x ∈ A | a < x < b}, (−∞, b] = {x ∈ A | x ≤ b}

and similarly for other types of intervals, [a, b], (a, b], (−∞, b] etc.

If (a, b) = ∅ then a is the immediate predecessor of b, and b the immediate successor of a.

Let (A, <) be an ordered set and B ⊂ A a subset.

• M is a largest element of B if M ∈ B and b ≤ M for all b ∈ B. The element m is a smallest element of B if m ∈ B and m ≤ b for all b ∈ B. We denote the largest element (if it exists) by max B and the smallest element (if it exists) by min B.

• M is an upper bound for B if M ∈ A and b ≤ M for all b ∈ B. The element m is a lower bound for B if m ∈ A and m ≤ b for all b ∈ B. The set of upper bounds is ⋂_{b∈B}[b, ∞) and the set of lower bounds is ⋂_{b∈B}(−∞, b].

• If the set of upper bounds has a smallest element, min ⋂_{b∈B}[b, ∞), it is called the least upper bound for B and denoted sup B. If the set of lower bounds has a largest element, max ⋂_{b∈B}(−∞, b], it is called the greatest lower bound for B and denoted inf B.

Definition 1.15. An ordered set (A, <) has the least upper bound property if any nonempty subset of A that has an upper bound has a least upper bound. If also (x, y) ≠ ∅ for all x < y, then (A, <) is a linear continuum.

Example 1.16. (1) R and (0, 1) have the same order type. [0, 1) and (0, 1) have distinct order types for [0, 1) has a smallest element and (0, 1) doesn’t. {−1} ∪ (0, 1) and [0, 1) have the same order type as we all can find an explicit order isomorphism between them.

(2) R × R has a linear dictionary order. What are the intervals (1 × 2, 1 × 3), [1 × 2, 3 × 2] and (1 × 2, 3 × 4]? Is R × R a linear continuum? Is [0, 1] × [0, 1]?

(3) We now consider two subsets of R × R. The dictionary order on Z_{+} × [0, 1) has the same order type as [1, ∞) so it is a linear continuum. In the dictionary order on [0, 1) × Z_{+} each element (a, n) has (a, n + 1) as its immediate successor so it is not a linear continuum. Thus Z_{+} × [0, 1) and [0, 1) × Z_{+} do not have the same order type. (So, in general, (A, <) × (B, <) and (B, <) × (A, <) represent different order types. This is no surprise since the dictionary order is not symmetric in the two variables.)

(4) (R, <) is a linear continuum as we all learn in kindergarten. The sub-ordered set (Z_{+}, <) has the least upper bound property but it is not a linear continuum as (1, 2) = ∅.

(5) (−1, 1) has the least upper bound property: Let B be any bounded from above subset of (−1, 1) and let M ∈ (−1, 1) be an upper bound. Then B is also bounded from above in R, of course, so there is a least upper bound, sup B, in R. Now sup B is the smallest upper bound so that sup B ≤ M < −1. We conclude that sup B lies in (−1, 1) and so it is also a least upper bound in (−1, 1). In fact, any convex subset of a linear continuum is a linear continuum.

(6) R − \{0\} does not have the least upper bound property as the subset B = \{−1, −\frac{1}{2}, −\frac{1}{3}, …\} is bounded from above (by say 100) but the set of upper bounds (0, ∞) has no smallest element.

2. The integers and the real numbers

We shall assume that the real numbers R exists with all the usual properties: (R, +, ·) is a field, (R, +, ·, <) is an ordered field, (R, <) is a linear continuum (1.15).

What about Z_{+}?

Definition 2.1. A subset A ⊂ R is inductive if 1 ∈ A and a ∈ A ⟹ a + 1 ∈ A.

There are inductive subsets of R, for instance R itself and [1, ∞).

Definition 2.2. Z_{+} is the intersection of all inductive subsets of R.

We have that 1 ∈ Z_{+} and Z_{+} ⊂ [1, ∞) because [1, ∞) is inductive so 1 = min Z_{+} is the smallest element of Z_{+}.

Theorem 2.3. (Induction Principle) Let J be a subset of Z_{+} such that

1 ∈ J and ∀n ∈ Z_{+} : n ∈ J ⟹ n + 1 ∈ J

Then J = Z_{+}.

Proof. J is inductive so J contains the smallest inductive set, Z_{+}. □

Theorem 2.4. Any nonempty subset of Z_{+} has a smallest element.

Before the proof, we need a lemma.

For each n ∈ Z_{+}, write

S_{n} = {x ∈ Z_{+} | x < n}

for the set of positive integers smaller than n (the section below n). Note that S_{1} = ∅ and S_{n+1} = S_{n} ∪ {n}.

Lemma 2.5. For any n ∈ Z_{+}, any nonempty subset of S_{n} has a smallest element.

Proof. Let J ⊂ Z_{+} be the set of integers for which the lemma is true. It is enough (2.3) to show that J is inductive. 1 ∈ J for the trivial reason that there are no nonempty subsets of S_{1} = ∅. Suppose that n ∈ J. Consider a nonempty subset A of S_{n+1}. If A consists of n alone, then n = min A is the smallest element of A. If not, A contains integers < n, and then min(A ∩ S_{n}) is the smallest element of A. Thus n + 1 ∈ J. □

Proof of Theorem 2.4. Let A ⊂ Z_{+} be any nonempty subset. The intersection A ∩ S_{n} is nonempty for some n, so it has a smallest element (2.5). This is also the smallest element of A. □

Theorem 2.6 (General Induction Principle). Let J be a subset of Z_{+} such that

∀n ∈ Z_{+} : S_{n} ⊂ J ⟹ n ∈ J

Then J = Z_{+}.

Proof. We show the contrapositive. Let J be a proper subset of Z_{+}. Consider the smallest element n = min(Z_{+} − J) outside J. Then n ∉ J and S_{n} ⊂ J (for n is the smallest element not in J meaning that all elements smaller than n are in J). Thus J does not satisfy the hypothesis of the theorem. □

Theorem 2.7 (Archimedean Principle). Z_{+} has no upper bound in R: For any real number there is a natural number which is greater.

Proof. We assume the opposite and derive a contradiction. Suppose that Z_{+} is bounded from above. Let b = sup Z_{+} be the least upper bound (R has the least upper bound property). Since b − 1 is not an upper bound (it is smaller than the least upper bound), there is a positive integer n ∈ Z_{+} such that n > b − 1. Then n + 1 is also an integer (Z_{+} is inductive) and n + 1 > b. This contradicts that b is an upper bound for Z_{+}. □

Theorem 2.8 (Principle of Recursive Definitions). For any set B and any function

ρ: map({S_{n} | n ∈ Z_{+}}, B) → B

there exists a unique function h: Z_{+} → B such that h(n) = ρ(h|S_{n}) for all n ∈ Z_{+}.

Proof. See [8, Ex 8.8]. □

This follows from the Induction Principle, but we shall not go into details. It is usually considered bad taste to define h in terms of h but the Principle of Recursive Definition is a permit to do exactly that in certain situations. Here is an example of a recursive definition from computer programing

 fibo := func< n | n le 2 select 1 else $$(n-1) + $$(n-2) >;

of the Fibonacci function. Mathematicians (sometimes) prefer instead to apply the Principle of Recursive Definitions to the map

ρ(S_{n} ⟶^{f} Z_{+}) = \begin{cases}1 & n < 2 \\ f(n − 2) − f(n − 1) & n > 2 \end{cases}

Recursive functions can be computed by Turing machines.

[figure: ∐_{j∈J} A_{j} ⊂ J × ⋃_{j∈J} A_{j}; J; ⋃_{j∈J} A_{j}]

Figure 1. The coproduct

3. Products and coproducts

Definition 3.1. An indexed family of sets consists of a collection A of sets, an index set J, and a surjective function f: J → A.

We often denote the set f(j) by A_{j} and the whole indexed family by {A_{j}}_{j∈J}. Any collection A can be made into an indexed family by using the identity map A → A as the indexing function.

We define the union, the intersection, the product, and the coproduct of the indexed family as

⋂_{j∈j} A_{j} = {a | a ∈ A_{j} for all j ∈ J},  ⋃_{j∈j} A_{j} = {a | a ∈ A_{j} for at least one j ∈ J}
∏_{j∈J} A_{j} = {x ∈ map(J, ⋃ A_{j}) | ∀j ∈ J: x(j) ∈ A_{j}} ∐_{j∈J} A_{j} = ⋃_{j∈J}{(j, a) ∈ J × ⋃_{j∈J} A_{j} | a ∈ A_{j}}

There are natural maps

\xymatrix{∐ A_{j} \ar[r]^{A_{j}→j} & J \ar[l]^{∏ A_{j}}}
π_{j}: ∏_{j∈J} A_{j} → A_{j} (projection) ι_{j}: A_{j} → ∐_{j∈J} A_{j} (injection)

given by π_{j}(x) = x(j) and ι_{j}(a) = (j, a) for all j ∈ J. These maps are used in establishing the identities

map(X, ∏_{j∈J} A_{j}) = ∏_{j∈J} map(X, A_{j}), map(∐_{j∈J} A_{j}, Y) = ∏_{j∈J} map(A_{j}, Y)

for any sets X and Y. This gives in particular maps

Δ: ⋂_{j∈J} A_{j} → ∏_{j∈J} A_{j} (diagonal), ∇: ∐_{j∈J} A_{j} → ⋃_{j∈J} A_{j} (codiagonal)

If the index set J = S_{n+1} = {1, …, n} then we also write

A_{1} ∪ ⋯ ∪ A_{n}, A_{1} ∩ ⋯ ∩ A_{n}, A_{1} × ⋯ × A_{n} A_{1} ⨿ ⋯ ⨿ A_{n}

for ⋃_{j∈S_{n+1}} A_{j}, ⋂_{j∈S_{n+1}} A_{j}, ∏_{j∈S_{n+1}} A_{j} ∐_{j∈S_{n+1}} A_{j}, respectively. If also and A_{j} = A for all j ∈ S_{n+1} we write A^{n} for the product ∏_{j∈S_{n+1}} A. The elements of A^{n} are all n-tuples (a_{1}, …, a_{n}) of elements from A.

If the index set J = Z_{+} then we also write

A_{1} ∪ ⋯ ∪ A_{n} ∪ ⋯, A_{1} ∩ ⋯ ∩ A_{n} ∩ ⋯, A_{1} × ⋯ × A_{n} × ⋯ A_{1} ⨿ ⋯ ⨿ A_{n} × ⋯

for ⋃_{j∈Z_{+}} A_{j}, ⋂_{j∈Z_{+}} A_{j}, ∏_{j∈Z_{+}} A_{j}, ∐_{j∈Z_{+}} A_{j}, respectively. If also A_{j} = A for all j we write A^{ω} for the product ∏_{j∈Z_{+}} A, the set of all functions x: Z_{+} → A, i.e. all sequences (x_{1}, …, x_{n}, …) of elements from A. ^{2}

Example 3.2. (1) S_{1} ∪ S_{2} ∪ ⋯ ∪ S_{n} ⋯ = ⋃_{n∈Z_{+}} S_{n} = Z_{+}.

(2) If the collection A = {A} consists of just one set A then ⋂_{j∈J} A = A = ⋃_{j∈J} A, ∏_{j∈J} A = map(J, A), and ∐_{j∈J} A = J × A.

(3) There is a bijection (which one?) between {0, 1}^{ω} = map(Z_{+}, {0, 1}) and P(Z_{+}). More generally, there is a bijection (which one) between the product ∏_{j∈J}{0, 1} = map(J, {0, 1}) and the power set P(J).

2ω is the formal set within set theory corresponding to the naïve set Z_{+} [11, V.1.5]

Even though we shall not specify our (ZF) axioms for set theory, let us mention just one axiom which has a kind of contended status since some of its consequences are counter-intuitive.

[figure: •; •; •]

Axiom 3.3 (Axiom of Choice (AC)). For any nonempty colllection of nonempty disjoint sets, A, there exists a set C ⊂ ⋃_{A∈A} A such that C ∩ A contains exactly one element for all A ∈ A.

If the ZF axioms of set theory are consistent, then both ZF+AC (Gödel 1938) and ZF+¬AC (Fraenkel and Mostowsski, Cohen) are consistent theories [11, IV.2.8]. You may take or leave AC without penalty. (Just like you may take or leave Euclid’s axiom on parallels depending on what kind of geometry you like to do.) We shall here include AC and work within ZFC (ZF + AC).

Unlike the other axioms of set theory, the AC does not determine the set C uniquely.

Theorem 3.4. [3, Thm B.18] The following statements are equivalent:

(1) The Axiom of Choice

(2) Any surjective map has a right inverse.

(3) For any nonempty indexed family of (not necessarily disjoint) nonempty sets, {A_{j}}_{j∈J}, there exists a function c: J → ⋃_{j∈J} A_{j} (a choice function) such that c(j) ∈ A_{j} for all j ∈ J.

(4) ∏_{j∈J} A_{j} ≠ ∅ for any nonempty indexed family of nonempty sets.

Proof. (1) ⟹ (2): Let f: A → B be a surjective map. Define the right inverse g: B → A by {g(b)} = C ∩ f^{−1}(b) where C ⊂ A = ⋃_{b∈B} f^{−1}(b) is a set such that C ∩ f^{−1}(b) contains exactly one point for each b ∈ B.

(2) ⟹ (3): Define c to be J → ∐ A_{j} ⟶^{∇} ⋃ A_{j} where the first map is a right inverse to the function ∐_{j∈J} A_{j} → J taking A_{j} to j for all j ∈ J.

(3) ⟺ (4): The product is defined to be the set of choice functions.

(3) ⟹ (1): Let A be a nonempty collection of nonempty sets. Put C = c(A) where c: A → ⋃_{A∈A} A is a choice function. □

[figure: c: J → ∐_{j∈J} A_{j}; J; ⋃_{j∈J} A_{j}; •; •; •; •; •; •; •; •]

Figure 2. A choice function

Here is a special, but often used, case. Let A be any nonempty set and P′(A) = P(A) − {∅} the collection of nonempty subsets of A. Then there exists (3.4.(3)) a choice function c: P′(A) → A such that c(B) ∈ B for any nonempty B ⊂ A. (The choice function selects an element in each nonempty subset of A.)

4. Finite and infinite sets

Definition 4.1. A set A is finite if S_{n+1} ∼ A for some n ∈ Z_{+}. A set is infinite if it is not finite.

We write X ∼ Y if there is a bijection between the two sets X and Y.

Lemma 4.2. Let n ∈ Z_{+} and let B be a proper subset of S_{n+1}.

(1) It is impossible to map B onto S_{n+1}.

(2) S_{m+1} ∼ B for some m < n.

Proof. Both statements are proved by induction.

(1) If n = 1, then S_{2} = {1} and B = ∅ so the assertion is true in this case. Assume it is true for some n ∈ Z_{+}. Consider a proper subset B of S_{n+1+1}. Suppose that there exists a surjection f: B → S_{n+1+1}. By permuting the elements of B and S_{n+1+1} if necessary, we can assume that f(n + 1) = n + 1. Then B − f^{−1}(n + 1) is a proper subset of S_{n+1} which is mapped onto S_{n+1} by f. But that is impossible by induction hypothesis.

(2) If n = 1, then S_{2} = {1} and B = ∅ so S_{1} ∼ B. Assume the assertion is true for some n ∈ Z_{+}. Consider a proper subset B of S_{n+1+1}. By permuting the elements of S_{n+1+1} if necessary, we can assume that n + 1 ∉ B so that B is a subset of S_{n+1}. If B = S_{n+1}, then B ∼ S_{n+1}, of course. Otherwise, B is a proper subset of S_{n+1} and then S_{m+1} ∼ B for some m < n < n + 1 by induction hypothesis. □

Corollary 4.3. Let A be a finite set.

(1) If S_{m+1} ∼ A ∼ S_{n+1}, then m = n.

(2) Any subset of A is finite.

Proof. (1) Suppose that m ≠ n. We may then assume that m < n. But then S_{m+1} is a proper subset of S_{n+1} which can be mapped onto S_{n+1}. That is not possible.

(2) Since this is true for the model finite set S_{n+1}, it is true for all finite sets. □

We have just learned that if A is finite then S_{n+1} ∼ A for a unique n ∈ Z_{+}. This n is called the cardinality of A and it is denoted cardA or |A|. We also learned that if B ⊂ A then B is finite, |B| ≤ |A|, and

(4.4)  |B| < |A| ⟺ B ⊊ A

which is often called the ‘pidgeon-hole principle’.

Theorem 4.5 (Characterization of finite sets). Let A be a set. The following statements are equivalent

(1) A is finite

(2) There exists a surjection S_{n+1} → A for some n ∈ Z_{+}

(3) There exists an injection A → S_{n+1} for some n ∈ Z_{+}

Proof. (1) ⟹ (2): There even exists a bijection S_{n+1} → A.

(2) ⟺ (3): 1.4.(2)

(3) ⟹ (1): If there exists an injection A → S_{n+1}, then there exists a bijection between A and a subset of S_{n+1}. But we have just seen that all subsets finite sets are finite. □

Corollary 4.6 (Hereditary properties of finite sets).

(1) Subsets of finite sets are finite.

(2) Images of finite sets are finite.

(3) Finite unions of finite sets are finite.

(4) Finite Cartesian products of finite sets are finite.

Proof. (1) Proved already in 4.3.

(2) S_{n+1} ↠ A ↠ B.

(3) To see that the union of two finite sets is finite, it is enough to show S_{m+1}⨿S_{n+1} is finite (for the union of any two finite sets is an image of this set). But it is immediate that S_{m+n+1} ∼ S_{m+1}⨿S_{n+1}. Induction now shows that A_{1} ∪ ⋯ ∪ A_{n} is finite when A_{1}, …, A_{n} are finitely many finite sets.

(4) Let A and B be finite. Since A × B = ∐_{a∈A} B is the union of finitely many finite sets, it is finite. Induction now shows that A_{1} × ⋯ × A_{n} is finite when A_{1}, …, A_{n} are finitely many finite sets. □

Are all sets finite? No!

Corollary 4.7. Z_{+} is infinite.

Proof. There is a surjective map of the proper subset Z_{+} − {1} onto Z_{+}. □

Theorem 4.8 (Characterization of infinite sets). Let A be a set. The following are equivalent: (1) A is infinite (2) There exists an injective map Z_{+} → A

(3) There exists a surjective map A → Z_{+}

(4) A is in bijection with a proper subset of itself

Proof. (1) ⟹ (2): Let c: P′(A) → A be a choice function. Define h: Z_{+} → A recursively by

h(1) = c(A)
h(i) = c(A − {h(1), …, h(i − 1)}),  i > 1

Then h is injective (if i < j then h(j) ∈ A − {h(1), …, h(i), …, h(j − 1)} so h(i) ≠ h(j)).

(2) ⟺ (3): 1.4.(2)

(2) ⟹ (4): We view Z_{+} as a subset of A. Then A = (A − Z_{+}) ∪ Z_{+} is in bijection with the proper subset A − {1} = (A − Z_{+}) ∪ (Z_{+} − {1}).

(4) ⟹ (1): This is 4.2. □

Here we applied the Principle of Recursive Definitions (2.8) to ρ(S_{n} ⟶^{f} Z_{+}) = c(A − f(S_{n})).

5. Countable and uncountable sets

Definition 5.1. A set C is countably infinite if Z_{+} ∼ C. It is countable if it is finite or countably infinite. It is uncountable if it is not countable.

Lemma 5.2. Any subset of Z_{+} is either finite or countably infinite (in bijection with Z_{+}).

Proof. Let C ⊂ Z_{+} be an infinite set of positive integers. We show (the stronger statement) that C has the order type of Z_{+}. Define a function h: Z_{+} → C recursively (2.8) by

h(1) = min C
h(i) = min(C − {h(1), …, h(i − 1)}), i > 1

using 2.4. Note that C − {h(1), …, h(i − 1)} is nonempty since C is infinite (4.5). We claim that h is bijective. \underline{h is order preserving}: If i < j, then

h(i) = min(C − {h(1), …, h(i − 1)}) < min(C − {h(1), …, h(i − 1), …, h(j − 1)}) = h(j)

because C − {h(1), …, h(i − 1)} ⊋ C − {h(1), …, h(i − 1), …, h(j − 1)}. \underline{h is surjective}: Let c ∈ C. We must find a positive integer m such that c = h(m). Our only hope is

m = min{n ∈ Z_{+} | h(n) ≥ c}

(Note that this has a meaning since the set {n ∈ Z_{+} | h(n) ≥ c} is nonempty as we can not inject the infinite set Z_{+} into the finite set {1, …, c − 1} = S_{c} (4.3). Note also that again we use 2.4.) By definition of m,

h(m) ≥ c and h(n) ≥ c ⇒ n ≥ m

The last of these two properties is equivalent to n < m ⇒ h(n) < c, so c ∉ {h(1), …, h(m − 1)}, or c ∈ C − {h(1), …, h(m − 1)}, and therefore

h(m) = min(C − {h(1), …, h(m − 1)}) ≤ c

by definition of h. Thus h(m) = c. □

Here we applied the Principle of Recursive Definitions (2.8) to ρ(S_{n} ⟶^{f} C) = min(C − f(S_{n})).

Theorem 5.3 (Characterization of countable sets). Let A be a set. The following statements are equivalent

(1) A is countable

(2) There exists a surjection Z_{+} → A

(3) There exists an injection A → Z_{+}

Proof. If A is finite, the theorem is true, so we assume that A is countably infinite.

(1) ⟹ (2): Clear.

(2) ⟺ (3): 1.4.(2)

(3) ⟹ (1): We may as well assume that A ⊂ Z_{+}. Since A is assumed to infinite, A is countably infinite by Lemma 5.2. □

Example 5.4. Z_{+} is obviously infinitely countable. The map f: Z_{+} × Z_{+} → Z_{+} given by f(m, n) = 2^{m}3^{n} is injective by uniqueness of prime factorizations. The map g: Z_{+} × Z_{+} → Q_{+} given by g(m, n) = \frac{m}{n}, is surjective. Thus Z_{+} × Z_{+} and Q_{+} are infinitely countable.

Corollary 5.5 (Hereditary properties of countable sets).

(1) A subset of a countable set is countable

(2) The image of a countable set is countable.

(3) A countable union of countable sets is countable (assuming AC).^{3}

(4) A finite product of countable sets is countable.

Proof. (1) B ↣ A ↣ Z_{+}.

(2) Z_{+} ↠ A ↠ B.

(3) Let {A_{j}}_{j∈J} be an indexed family of sets where J is countable and each set A_{j} is countable. It is enough to show that ∐ A_{j} is countable. We leave the case where the index set J is finite as an exercise and consider only the case where J is infinite. Then we may as well assume that J = Z_{+}. Choose (!) for each n ∈ Z_{+} an injective map f_{n}: A_{n} → Z_{+}. Then we have injective maps

∐ A_{n} \underrightarrow{∐ f_{n}} ∐ Z_{+} = Z_{+} × Z_{+} ⟶^{(5.4)} Z_{+}

so ∐ A_{n} is countable.

(4) If A and B are countable, so is A × B = ∐_{a∈A} B as we have just seen. Now use induction to show that if A_{1}, …, A_{n} are countable, so is A_{1} × ⋯ × A_{n}. □

You may think that a countable product of countable sets is countable or indeed that all sets are finite or countable – but that’s false.

Theorem 5.6. Let A be any set.

(1) There is no injective map P(A) → A

(2) There is no surjective map A → P(A)

Proof. (Cantor’s diagonal argument.) Is is a general fact (1.4.(2)) that (1) ⟺ (2). Thus it suffices to show (2). Let g: A → P(A) be any function. Then

{a ∈ A | a ∉ g(a)} ∈ P(A)

is not in the image of g. Because if this set were of the form g(b) for some b ∈ A, then we’d have

b ∈ g(b) ⟺ b ∉ g(b) ⚡

□

Corollary 5.7. The set P(Z_{+}) = map(Z_{+}, {0, 1}) = ∏_{n∈Z_{+}}{0, 1} = {0, 1}^{ω} is uncountable.

Russel’s paradox also exploits Cantor’s diagonal argument.

We have seen (5.2) that any subset of Z_{+} is either finite or in bijection with Z_{+}. What about subsets of R?

Conjecture 5.8 (Cantor’s Continuum Hypothesis, CH). Any subset of R is either countable or in bijection with R.

CH is independent of the ZFC axioms for set theory in that if ZFC is consistent then both ZFC+CH (Gödel 1950) and ZFC+¬CH (Cohen 1963) are consistent theories [11, VII.4.26] [4]. Our axioms are not adequate to settle the CH.

Look up the generalized continuum hypothesis (GCH) [Ex 11.8] (due to Hausdorff) somewhere [14, 15]. It is not customary to assume the GCH; if you do, the AC becomes a theorem.

3In set theory without AC, R is a countable union of countable sets [11, p 228]

6. Well-ordered sets

We have seen that all nonempty subsets of (Z_{+}, <) have a smallest element and we have used this property in quite a few places so there is reason to suspect that this is an important property in general. This is the reason for the following definition. You may think of well-ordered sets as some kind of generalized versions of Z_{+}.

Definition 6.1. A set A with a linear order < is well-ordered if any nonempty subset has a smallest element.

Any well-ordered set has a smallest element. Any element (but the largest) in a well-ordered set has an immediate successor, the smallest successor [Ex 10.2]. (And any element (but the smallest) has an immediate predecessor?) A well-ordered set can not contain an infinite descending chain x_{1} > x_{2} > ⋯, in fact, a linearly ordered set is well-ordered if and only if it does not contain a copy of the negative integers Z_{−} [Ex 10.4].

Let (A, <) be a well-ordered set and α an element of A. The subset

S_{α}(A) = S_{α} = (−∞, α) = {a ∈ A | a < α}

is called the section of A by α.

The induction principle and the principle of recursive defintions apply not only to Z_{+} but to any well-ordered set.

Theorem 6.2 (Principle of Transfinite Induction). [Ex 10.7] (Cf 2.6) Let (A, <) be a well-ordered set and J ⊂ A a subset such that

∀α ∈ A: S_{α} ⊂ J ⟹ α ∈ J

Then J = A.

Proof. Formally identical to the proof of 2.6. □

Theorem 6.3 (Principle of Transfinite Recursive Definitions). Let (A, <) be a well-ordered set. For any set B and any function

ρ: map({S_{α} | α ∈ A}, B) → B

there exists a unique function h: A → B such that h(α) = ρ(h|S_{α}) for all α ∈ A.

Proposition 6.4 (Hereditary properties of well-ordered sets).

(1) A subset of a well-ordered set is well-ordered.

(2) The coproduct of any well-ordered family of well-ordered sets is well-ordered [Ex 10.8].

(3) The product of any finite family of well-ordered sets is well-ordered.

Proof. (1) Clear.

(2) Let J be a well-ordered set and {A_{j}}_{j∈J} a family of well-ordered sets indexed by J. For i, j ∈ J and x ∈ A_{i}, y ∈ A_{j}, define

(i, x) < (j, y) ⟺^{def} i < j or (i = j and x < y)

and convince yourself that this is a well-ordering.

(3) If (A, <) and (B, <) well-ordered then A × B = ∐_{a∈A} B is well-ordered. Now use induction to show that the product A_{1} × ⋯ × A_{n} of finitely many well-ordered sets A_{1}, …, A_{n} is wellordered.

□

If C a nonempty subset of A × B then min C = (c_{1}, min π_{2}(C ∩ π_{1}^{−1}(c_{1}))) where c_{1} = min π_{1}(C) is the smallest element of C.

Example 6.5. (1) The positive integers (Z_{+}, <) is a well-ordered set.

(2) Z and R are not well-ordered sets.

(3) Any section S_{n+1} = {1, 2, …, n} of Z_{+} is well-ordered (6.4.(1)).

(4) S_{n+1} = {1, 2, …, n} of Z_{+} is well-ordered. The product S_{n+1} × Z_{+} is well-ordered (6.4.(3)). The finite products Z_{+}^{n} = Z_{+} × Z_{+} ⋯ × Z_{+} are well-ordered (6.4.(3)).

(5) The infinite product {0, 1}^{ω} is not well-ordered for it contains the infinite descending chain (1, 0, 0, 0, …) > (0, 1, 0, 0, …) > (0, 0, 1, 0, …) > · · ·.

(6) The set \overline{S}_{ω} = [1, ω] = Z_{+} ⨿ \{ω\} is well-ordered (6.4.(2)). It has ω as its largest element. The section S_{ω} = [1, ω) = Z_{+} is countably infinite but any other section is finite. Any finite subset A of [1, ω) has an upper bound because the set of non-upper bounds

{x ∈ [1, ω) | ∃a ∈ A: x < a} = ⋃_{a∈A} S_{a}

is finite (4.6.(3)) but [1, ω) is infinite. \overline{S}_{ω} has the same order type as the interval [1 × 1, 2 × 1] in Z_{+} × Z_{+}. Which of these well-ordereds have the same order type [Ex 10.3]? Draw pictures of examples of

well-ordered sets.

We can classify completely all finte well-ordered sets.

Theorem 6.6 (Finite order types). [Ex 6.4] Any finite linearly ordered set A of cardinality n has the order type of (S_{n+1}, <); in particular, it is well-ordered and it has a largest element.

Proof. Define h: S_{n+1} → A recursively by h(1) = min A and

h(i) = min(A − {h(1), …, h(i − 1)}, i > 1

Then h is order preserving. In particular, h is injective and hence bijective (by the pidgeon hole principle (4.4)) since the two sets have the same cadinality. □

Can you find an explicit order preserving bijection S_{m+1} × S_{n+1} → S_{m+n+1}?

So there is just one order type of a given finite cardinality n. There are many countably infinite well-ordered sets (6.5). Is there an uncountable well-ordered set? Our examples, R and {0, 1}^{ω}, of uncountable sets are not well-ordered (6.5.(2), 6.5.(5)).

Theorem 6.7 (Well-ordering theorem). (Zermelo 1904) Any set can be well-ordered.

We focus on the minimal criminal, the minimal uncountable well-ordered set. (It may help to look at 6.5.(6) again.)

Lemma 6.8. There exists a well-ordered set \overline{S}_{Ω} = [0, Ω] with a smallest element, 0, and a largest element, Ω, such that:

(1) The section S_{Ω} = [0, Ω) by Ω is uncountable but any other section, S_{α} = [0, α) for α < Ω, is countable.

(2) Any countable subset of S_{Ω} = [0, Ω) has an upper bound in S_{Ω} = [0, Ω)

Proof. (Cf 6.5.(6)) Take any uncountable well-ordered set A. Append a greatest element to A. Call the result A again. Now A has at least one uncountable section. Let Ω be the smallest element of A such that the section by this element is uncountable, that is Ω = min{α ∈ A | S_{α} is uncountable}. Put \overline{S}_{Ω} = [0, Ω] where 0 is the smallest element of A. This well-ordered set satisfies (1) and (2). Let C be a countable subset of S_{Ω} = [0, Ω). We want to show that it has an upper bound. We consider the set of elements of S_{Ω} that are not upper bounds, i.e

\{x ∈ S_{Ω} | x is not an upper bound for C\} = \{x ∈ S_{Ω} | ∃c ∈ C: x < c\} = \overbrace{⋃_{c∈C} S_{c}}^{countable} ⊊ \overbrace{S_{Ω}}^{uncountable}

This set of not upper bounds is countable for it is a countable union of countable sets (5.5.(4)). But S_{Ω} is uncountable, so the set of not upper bounds is a proper subset. □

See [SupplExI : 8] for an explicit construction of a well-ordered uncountable set. Z_{+} is the well-ordered set of all finite (nonzero) order types and S_{Ω} is the well-ordered set of all countable (nonzero) order types. See [Ex 10.6] for further properties of S_{Ω}.

Recall that the ordered set Z_{+}×[0, 1) is a linear continuum of the same order type as [1, ∞) ⊂ R. What happens if we replace Z_{+} by S_{Ω} [Ex 24.6, 24.12]?

7. Partially ordered sets, The Maximum Principle and Zorn’s lemma

If we do not insist on comparability in our order relation we obtain a partially ordered set (poset):

Definition 7.1. A strict partial order on a set A is a relation ≺ on A which is non-reflexive and transitive:

(1) a ≺ a holds for no a ∈ A

(2) a ≺ b and b ≺ c implies a ≺ c.

We do not require that any two nonidentical element can be compared. For instance, power sets are strictly partially ordered by proper inclusion ⊊ (in fact, it is good idea to read ≺ as ”is contained in”).

Theorem 7.2 (Hausdorff’s Maximum Principle). Any linearly ordered subset of a poset is contained in a maximal linearly ordered subset.

Proof. We prove that any poset contains maximal linearly ordered subsets and leave general form of the theorem to the reader. As a special case, suppose that the poset is infinitely countable. We may as well assume that the poset is Z_{+} with some partial order ≺. Define h: Z_{+} → {0, 1} recursively by h(1) = 0 and

h(i) = \begin{cases}1 & \{j < i|h(j) = 0\} ∪ \{i\} is linearly ordered wrt ≺ \\ 0 & otherwise \end{cases}

for i > 0. Then H = h^{−1}(0) is a maximal linearly ordered subset.

□

For the proof in the general case, let (A, <) be a poset, well-order A (6.7), apply the Principle of Transfinite Recursion (6.3) to the function

ρ(S_{α} ⟶^{f} \{0, 1\}) = \begin{cases}0 & f^{−1}(0) ∪ \{α\} is a linearly ordered subset of A \\ 1 & otherwise \end{cases}

and get a function h: A → {0, 1}.

Definition 7.3. Let (A, ≺) be a set with a strict partial order. An element m of A is maximal if no elements of A are greater than m.

Let B be a subset of A. If b ⪯ c for all b ∈ B, then c is an upper bound on B.

Maximal elements are often of great interest.

Theorem 7.4 (Zorn’s lemma). Let (A, <) be a poset. Suppose that any linearly ordered subset of A has an upper bound. Then A contains maximal elements.

Proof. Any upper bound on a maximal linearly ordered subset is maximal: Let H ⊂ A be a maximal linearly ordered subset. By hypothesis, H has an upper bound m, i.e. x ⪯ m for all x ∈ H. By maximality of H, m must be in H, and there can be no element greater than m. (Suppose that m ≺ d for some d. Then x ⪯ m ≺ d for all elements of H so H ∪ {d} is linearly ordered, contradicting maximality of H.) □

We shall later use Zorn’s lemma to prove Tychonoff’s theorem (18.16) that a product of compact spaces is compact. In fact, The Axiom of Choice, Zermelo’s Well-ordering theorem, Hausdorff’s maximum principle, Zorn’s lemma, and Tychonoff’s theorem are equivalent.

Here are two typical applications. Recall that a basis for a vector space over a field is a maximal independent subset.

Theorem 7.5. [Ex 11.8] Any linearly independent subset of a vector space is contained in a basis.

Proof. Let M ⊂ V be a linearly independent subset. Apply Zorn’s lemma to the set, strictly partially ordered by proper inclusion ⊊, of independent subsets containing M. A maximal element is a basis. □

As a corollary we see that R and R^{2} are isomorphic as vector spaces over Q.

A maximal ideal in a ring is a maximal proper ideal.

englishGENERAL TOPOLOGY 17

Theorem 7.6. Any proper ideal of a ring is contained in a maximal ideal. Proof. Let I be a proper ideal. Consider the set, strictly partially ordered by proper inclusion, of proper ideals containing the given ideal I. □

Other authors prefer to work with partial orders instead of strict partial orders. Definition 7.7. [Ex 11.2] Let A be a set. A relation ⪯ on A is said to be a partial order precisely when it is symmetric (that is a ⪯ a for all a in A), transitive (that is a ⪯ b and b ⪯ c implies a ⪯ c), and anti-symmetric (that is a ⪯ b and b ⪯ a implies a = b).

8. Topological spaces

What does it mean that a map f: X → Y between two sets is continuous? To answer this question, and much more, we equip sets with topologies.

Definition 8.1. Let X be a set. A topology on X is a collection T of subsets of X, called open sets, such that

(1) ∅ and X are open

(2) The intersection of finitely many (two) open sets is open

(3) The union of an arbitrary collection of open sets is open A topological space is a set X together with a topology T on X.

Example 8.2. (1) In he trivial topology T = {∅, X}, only two subsets are open. (2) In the discrete topology T = P(X), all subsets are open.

(3) In the particular point topology, the open sets are ∅, X and all subsets containing a particular point x ∈ X. For instance, the Sierpinski space is the set X = {0, 1} with the particular point topology for the point 0. The open sets are T = {∅, {0}, X}.

(4) In the finite complement topology (or cofinite topology), the open sets are ∅ and X and all subsets with a finite complement. .

(5) The standard topology on the real line R is T = {unions of open intervals}.

(6) More generally, suppose that (X, d) is a metric space. The open r-ball centered at x ∈ X is the set B_{d}(x, r) = {y ∈ X | d(x, y) < r} of points within distance r > 0 from x. The metric topology on X is the collection T_{d} = {unions of open balls}. The open sets of the topological space (X, T_{d}) and the open sets in the metric space (X, d) are the same. (See §§20–21 for more on metric topologies.) The Sierpinski topology and the finite complement topology on an infinite set are not metric

topologies.

Topologies on X are partially ordered by inclusion. For instance, the finite complement topology (8.2.(4)) on R is contained in the standard topology (8.2.(5), and the indiscrete topology (8.2.(1)) on {0, 1} is contained in the Sierpinski topology (8.2.(3)) is contained in the discrete topology (8.2.(2)).

Definition 8.3 (Comparison of topologies). Let T and T′ be two topologies on the same set X.

\begin{matrix}T is finer than T′ \\ T′ is coarser than T \end{matrix} \} ⟺^{def} T ⊃ T′

The finest topology is the topology with the most opens sets, the coarsest topology is the one with fewest open sets. (Think of sandpaper!) The discrete topology is finer and the indiscrete topology coarser than any other topology: P(X) ⊃ T ⊃ {∅, X}. Of course, two topologies may also be incomparable.

8.4. Subbasis and basis for a topology. Any intersection of topologies is a topology [Ex 13.4]. A standard way of defining a topology is to specify some collection of subsets and look at the coarsest topology containing this collection. In praxis, we restrict ourselves to collections of subsets that cover X.

Definition 8.5. A basis on X is a collection, B, of subsets of X (called basis sets) such that (1) The basis sets cover X, ie X = ⋃_{B∈B} B

(2) The intersection of any two basis sets is a union of basis sets A subbasis is a collection S of subsets of X (called subbasis sets) that cover X.

If B ⊂ P(X) is a basis then

T_{B} = {unions of basis sets}

is a topology, called the topology generated by the basis B. It is the coarsests topology in which the basis sets are open [Ex 13.5]. See 8.2.(5) and 8.2.(3) for examples.

If S ⊂ P(X) is a subbasis then

T_{S} = {unions of finite intersections of subbasis sets} is a topology, called the topology generated by the subbasis S. It is the coarsests topology in which the subbasis sets are open [Ex 13.5]. (Use the distributive laws [§1] for ∪ and ∩.)

If S is a subbasis, then

(8.6)  B_{S} = {Finite intersections of S-sets}

is a basis generating the same topology as S, T_{B_{S}} = T_{S}.

A topology is a basis is a subbasis.

Lemma 8.7. Let T be a topology on X and B a collection of open subsets of X. If any open set is a union of sets from B then B is a basis for the topology T .

Proof. B is clearly a basis and B ⊂ T so that the topology generated by B is coarser than T . The second condition gives the opposite inclusion. □

How can we compare topologies given by bases? How can we tell if two bases, or a subbasis and a basis, generate the same topology? (Two topologies, bases or subbases are said to be equivalent if they generate the same topology.)

Lemma 8.8 (Comparison). Let B and B′ be two bases and S a subbasis.

(1)

T_{B} ⊂ T_{B′} ⟺ B ⊂ T_{B′} ⟺ All B-sets are open in T_{B′}

(2)

T_{B} = T_{B′} ⟺ \{ \begin{matrix}All B-sets are open in T_{B′} \\ All B′-sets are open in T_{B} \end{matrix}\}

(3)

T_{B} = T_{S} ⟺ \{ \begin{matrix}All B-sets are open in T_{S} \\ All S-sets are open in T_{B} \end{matrix}\}

Proof. (1) is obvious since T_{B} is the coarsest topology containing B. Item (2) is immediate from (1). Item (3) is proved in the same way since T_{B} = T_{S} iff T_{B} ⊂ T_{S} and T_{B} ⊃ T_{S} iff B ⊂ T_{S} and T_{B} ⊃ S. □

Example 8.9. (1) In a metric space, the set B = {B(x, r) | x ∈ X, r > 0} of open balls is (by definition) a basis for the metric topology T_{d}. The collection of open balls of radius \frac{1}{n}, n ∈ Z_{+}, is an equivalent topology basis for T_{d}.

(2) The collection of rectangular regions (a_{1}, b_{1}) × (a_{2}, b_{2}) in the plane R^{2} is a topology basis equivalent to the standard basis of open balls B(a, r) = {x ∈ R^{2} | |x − a| < r}. you can always put a ball inside a rectangle and a rectangle inside a ball.

(3) Let f: X → Y be any map. If T is a topology on Y with basis B or subbasis S , then the pull-back f^{−1}(T ) is a topology, the initial topology for f, on X with basis f^{−1}(B) and subbasis f^{−1}(S).

(4) More generally, let X be a set, {Y_{j}} a collection of topological spaces, and f_{j}: X → Y_{j}, j ∈ J, a set of maps. Let T_{j} be the topology on Y_{j}, B_{j} a basis and S_{j}, j ∈ J a subbasis. Then ⋃ f_{j}^{−1}(T_{j}), ⋃ f_{j}^{−1}(B_{j}), ⋃ f_{j}^{−1}(S_{j}) are equivalent subbases on X. The topology they generate is called the initial topology for the maps f_{j}, j ∈ J.

Example 8.10. [Topologies on R] We consider three topologies on R:

R: The standard topology with basis the open intervals (a, b).

R_{ℓ}: The right half-open interval topology with basis the right half-open intervals [a, b). R_{K}: The K-topology with basis \{(a, b)\} ∪ \{(a, b) − K\} where K = \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, …\}.

The right half-open interval topology is strictly finer than the standard topology because any open interval is a union of half-open intervals but not conversely (8.8) ((a, b) = ⋃_{a<x<b}[x, b), and [0, 1) is open in R_{ℓ} but not in R; an open interval containing 0 is not a subset of [0, 1)). The K-topology is strictly finer than the standard topology because its basis contains the standard basis and R−K is open in R_{K} but not in R (an open interval containing 0 is not a subset of R − K). The topologies R_{ℓ} and R_{K} are not comparable [Ex 13.6].

Example 8.11. The collection of all open rays

S = {(−∞, b)} ∪ {(a, +∞)}

is a subbasis and the collection B = {(a, b)} of all open intervals is a basis for the standard topology on R (8.10) by 8.8.(3).

9. Order topologies

We associate a topological space to any linearly ordered set and obtain a large supply of examples of topological spaces. You may view the topological space as a means to study the ordered set, to find invariants, or you may view this construction as a provider of interesting examples of topological spaces.

Let (X, <) be a linearly ordered set containing at least two points. The open rays in X are the subsets

(−∞, b) = {x ∈ X | x < b}, (a, +∞) = {x ∈ X | a < x}

of X. The collection of all open rays is clearly a subbasis (just as in 8.11).

Definition 9.1. The order topology T_{<} on the linearly ordered set X is the topology with all open rays as subbasis. A linearly ordered space is a linearly ordered set with the order topology.

The open intervals in X are the subsets of the form

(a, b) = (−∞, b) ∩ (a, +∞) = {x ∈ X | a < x < b}, a, b ∈ X, a < b.

Lemma 9.2. The collection of all open rays together with all open intervals is a basis for the order topology T_{<}.

Proof. B_{S} =^{(8.6)} {Finite intersections of S-sets} = S ∪ {(a, b)}. □

If X has a smallest element a_{0} then (−∞, b) = [a_{0}, b) is open. If X has no smallest element, then the open ray (−∞, b) = ⋃_{a<c}(a, c) is a union of open intervals and we do not need this open ray in the basis. Similar remarks apply to the greatest element when it exists.

Lemma 9.3. (Cf 8.11) If X has no smallest and no largest element, then the set {(a, b)} of open intervals is a basis for the order topology.

Example 9.4. (1) The order topology on the ordered set (R, <) is the standard topology (8.10). (2) The order topology on the ordered set R^{2} has as basis the collection of all open intervals (a_{1} × a_{2}, b_{1} × b_{2}). An equivalent basis (8.8) consists of the open intervals (a × b_{1}, a × b_{2}). R_{<}^{2} is strictly finer than R_{d}^{2}.

(3) The order topology on Z_{+} is the discrete topology because (−∞, n) ∩ (∞, n + 1) = {n} is open.

(4) The order topology on Z_{+} × Z_{+} is not discrete. Any open set that contains the element 2 × 1 also contains elements from {1} × Z_{+}. Thus the set {1 × 2} is not open.

(5) The order topology on Z × Z is discrete.

(6) Is the order topology on S_{Ω} discrete?

(7) I^{2} = [0, 1]^{2} with the order topology is denoted I_{o}^{2} and called the ordered square. The open sets containing the point x × y ∈ I_{o}^{2} look quite different depending on whether y ∈ {0, 1} or 0 < y < 1.

10. The product topology

Let (X_{j})_{j∈J} be an indexed family of topological spaces. Let π_{k}: ∏_{j∈J} X_{j} → X_{k} be the projection map. An open cylinder is a subset of the product space of the form

π_{k}^{−1}(U_{k}), U_{k} ⊂ X_{k} open, k ∈ J.

The set π_{k}^{−1}(U_{k}) consists of the points (x_{j}) ∈ ∏ X_{j} with kth coordinate in U_{k}. Alternatively, π_{k}^{−1}(U_{k}) consists of all choice functions c: J → ⋃_{j∈J} U_{j} such that c(k) ∈ U_{k} (Figure 2).

Definition 10.1. The product topology on ∏_{j∈J} X_{j} is the topology with subbasis

S∏ = ⋃_{j∈j}{π_{j}^{−1}(U) | U ⊂ X_{j} open}

consisting of all open cylinders or, equivalently, with basis (8.6)

B∏ = {∏_{j∈J} U_{j} | U_{j} ⊂ X_{j} open and U_{j} = X_{j} for all but finitely many j ∈ J}

The product topology is the coarsest topology making all the projection maps π_{j}: ∏ X_{j} → X_{j}, j ∈ J, continuous.

This becomes particularly simple when we consider finite products.

Lemma 10.2. Let X = X_{1} × X_{2} × ⋯ X_{k} be a finite Cartesian product. The collection

B = {U_{1} × U_{2} × ⋯ × U_{k} | U_{1} open in X_{1}, U_{2} open in X_{2}, …, U_{k} open in X_{k}}

of all products of open sets is a basis for the product topology.

Corollary 10.3. Suppose that B_{j} is a basis for the topology on X_{j}, j = 1, …, k. Then B_{1}×⋯×B_{k} is a basis for the product topology on X_{1} × ⋯ × X_{k}.

Proof. Note that B_{1} × ⋯ × B_{k} is indeed a basis. Compare it to the basis of 10.2 using 8.8. □

10.4. Products of linearly ordered spaces. When (X, <) and (Y, <) are linearly ordered sets, we now have two topologies on the Cartesian product X × Y: The product topology of the order topologies, (X, T_{<}) × (Y, T_{<}), and the order topology of the product dictionary order, (X × Y, T_{<}). These two topologies are in general not identical or even comparable. It is difficult to imagine any general relation between them since X_{<} × Y_{<} is essentially symmetric in X and Y whereas the dictionary order has no such symmetry. But even when X = Y there does not seem to be a general pattern: The order topology (Z_{+}×Z_{+}, T_{<}) is coarser than the product topology (Z_{+}, T_{<})×(Z_{+}, T_{<}) (which is discrete) (9.4.(4)–(5)). On the other hand, the order topology (R × R, T_{<}) is finer than the product topology (R, T_{<}) × (R, T_{<}) which is the standard topology, see 9.4.(1)–(2) and [Ex 16.9].

Corollary 10.5 (Cf [Ex 16.9]). Let (X, <) and (Y, <) be two linearly ordered sets. Suppose that Y does not have a largest or a smallest element. Then the order topology (X × Y)_{<} = X_{d} × Y_{<} where X_{d} is X with the discrete topology. This topology is finer than X_{<} × Y_{<}.

Proof. The equations

(−∞, a × b) = ⋃_{x<a}{x} × Y ∪ {a} × (−∞, b), (a × b, +∞) = {a} × (b, +∞) ∪ ⋃_{x>a}{x} × Y

show that the order topology is coarser than the product of the discrete and the order topology. On the other hand, the sets {x}×(c, d) is a basis for = X_{d}×Y_{<} (9.3, 10.3) and {x}×(c, d) = (x×c, x×d) is open in the order topology since it is an open interval. □

The corollary shows that the order topology on X × Y does not yield anything new in case the second factor is unbounded in both directions. As an example where this is not the case we have already seen Z_{+} × Z_{+} and I_{o}^{2} (9.4.(4), 9.4.(7)).

10.6. The coproduct topology. The coproduct topology on the coproduct ∐ X_{j} (3) is the finest topology making all the inclusion maps ι_{j}: X_{j} → ∐ X_{j}, j ∈ J, continuous.

This means that

U ⊂ ∐_{j∈J} X_{j} is open ⟺ ι_{j}^{−1}(U) is open for all j ∈ J

Alternatively, the open sets of the coproduct ∐ X_{j} are the coproducts ∐ U_{j} of open set U_{j} ⊂ X_{j}.

11. The subspace topology

Let (X, T ) be a topological space and Y ⊂ X a subset. We make Y into a topological space.

Definition 11.1. The subspace topology on Y is the topology T_{⊂} = Y ∩ T = {Y ∩ U | U ∈ T }. A subset V ⊂ Y is open relative to Y if it is open in the subspace topology.

It is immediate that T_{⊂} is indeed a topology. The subset V ⊂ Y is open relative to Y if and only if V = Y ∩ U for some open set U ⊂ X.

Lemma 11.2. If B is a basis for T , then Y ∩ B = {Y ∩ U | U ∈ B} is a basis for the subspace topology Y ∩ T . If S is a subbasis for T , then Y ∩ S = {Y ∩ U | U ∈ S} is a subbasis for the subspace topology Y ∩ T .

Proof. This is 8.9.(3) applied to the inclusion map A ↪ X. □

If V ⊂ Y is open, then V is also relatively open. The converse holds if Y is open.

Lemma 11.3. Assume that A ⊂ Y ⊂ X. Then

(1) A is \begin{cases}open \\ closed \end{cases}\} in Y ⟺ A = Y ∩ U for some \begin{cases}open \\ closed \end{cases}\} set U in X
(2) If Y is \begin{cases}open \\ closed \end{cases}\} then: A is \begin{cases}open \\ closed \end{cases}\} in Y ⟺ A is \begin{cases}open \\ closed \end{cases}\} in X

Proof. (1) This is the definition of the subspace topology.

(2) Suppose that Y is open and that A ⊂ Y. Then

A open in Y ⟺ A = Y ∩ U for some open U ⊂ X ⟺ A open in X

in that A = A ∩ Y. □

The lemma says that an open subset of an open subset is open and that a closed subset of a closed subset is closed.

The next theorem says that the subspace and the product space operations commute.

Theorem 11.4. Let Y_{j} ⊂ X_{j}, j ∈ J. The subspace topology that ∏ Y_{j} inherits from ∏ X_{j} is the product topology of the subspace topologies on Y_{j}.

Proof. The subspace topology on ∏ Y_{j} has subbasis

∏ Y_{j} ∩ S_{∏ X_{j}} = ∏ Y_{j} ∩ ⋃_{k∈j}{π_{k}^{−1}(U_{k})} = ⋃_{k∈J}{∏ Y_{j} ∩ π_{k}^{−1}(U_{k})}, U_{k} ⊂ X_{k} open,

and the product topology on ∏ Y_{j} has subbasis

S_{∏ Y_{j}} = ⋃_{k∈J}{π_{k}^{−1}(Y_{k} ∩ U_{k})}, U_{k} ⊂ X_{k} open

These two subbases are identical for π_{k}^{−1}(Y_{k} ∩U_{k}) =^{[8, Ex 2.2]} π_{k}^{−1}(Y_{k})∩π_{k}^{−1}(U_{k}) = (∏ Y_{j})∩π_{k}^{−1}(U_{k}). □

11.5. Subspaces of linearly ordered spaces. When (X, <) is a linearly ordered set and Y ⊂ X a subset we now have two topologies on Y. We can view Y as a subspace of the topological space X_{<} with the order topology or we can view Y as a sub-ordered set of X and give Y the order topology. Note that these two topologies are not the same when X = R and Y = [0, 1) ∪ {2} ⊂ R: In the subspace topology {2} is open in Y but {2} is not open in the order topology because Y has the order type of [0, 1]. See 11.7 for more examples. The point is that

• Any open ray in Y is the intersection of Y with an open ray in X

• The intersection of Y with an open ray in X need not be an open ray in Y However, if Y happens to be convex then open rays in Y are precisely Y intersected with open rays in X.

Lemma 11.6 (Y_{<} ⊂ Y ∩ X_{<}). Let (X, <) be a linearly ordered set and Y ⊂ X a subset. The order topology on Y is coarser than the subspace topology Y in general. If Y is convex, the two topologies on Y are identical.

Proof. The order topology on Y has subbasis

S_{<} = {Y ∩ (−∞, b) | b ∈ Y } ∪ {Y ∩ (a, ∞) | a ∈ Y }

and the subspace topology on Y has subbasis

S_{⊂} = {Y ∩ (−∞, b) | b ∈ X} ∪ {Y ∩ (a, ∞) | a ∈ X}

Clearly, S_{<} ⊂ S_{⊂}, so the order topology on Y is coarser than the subspace topology in general.

If Y is convex and b ∈ X − Y then b is either a lower or an upper bound for Y as we cannot have y_{1} < b < y_{2} for two points y_{1} and y_{2} of Y. If b is a lower bound for Y, then Y ∩ (−∞, b) = ∅, and if b is an upper bound for Y, then Y ∩ (−∞, b) = Y. Therefore also S_{⊂} ⊂ T_{<} so that in fact T_{<} = T_{⊂}.

□

Example 11.7. (1) \overline{S}_{2×1} = 1 × Z_{+} ∪ \{2 × 1\} = [1 × 1, 2 × 1] is a convex subset of Z_{+} × Z_{+}. The subspace topology (9.4.(4)) is the same as the order topology.

(2) The subset Z_{+} ×Z_{+} is not convex in Z×Z so we expect the subspace topology to be strictly finer than the order topology. Indeed, the subspace topology that Z_{+} × Z_{+} inherits from the discrete space Z × Z is discrete but the order topology is not discrete (9.4.(4)–(5)).

(3) Consider X = R^{2} and Y = [0, 1]^{2} with order topologies. Y is not convex so we expect the subspace topology to be strictly finer than the order topology. Indeed, the subspace topology on [0, 1]^{2}, which is [0, 1]_{d} × [0, 1], is strictly finer than I_{o}^{2} (9.4.(7)): The set

[0, \frac{1}{2}] × [0, 1] = ([0, 1] × [0, 1]) ∩ (−∞, \frac{1}{2} × 2)

is open in the subspace topology on Y but it is not open in the order topology on Y as any basis open set (9.2) containing \frac{1}{2} × 1 also contains points with first coordinate > \frac{1}{2}.

(4) Consider X = R^{2} and Y = (0, 1)^{2} with order topologies. Y is not convex so we expect the subspace topology to be strictly finer than the order topology. But it isn’t! The reason is that (0, 1) does not have a greatest nor a smallest element (10.5).

(5) The subset Q of the linearly ordered set R is not convex but nevertheless the subspace topology inherited from R is the order topology. Again, the reason seems to be that Q does not have a greatest nor a smallest element.

12. Closed sets and limit points

Let (X, T ) be a topological space and A a subset of X.

Definition 12.1. We say that the set A is closed if its complement X − A is open.

Note that X and ∅ are closed (and open), a finite union of closed sets is closed, an arbitrary intersection of closed sets is closed. The finer the topology, the more closed sets.

Example 12.2. (1) [a, b] is closed in R, [a, b) is neither closed nor open, R is both closed and open.

(2) Let X = [0, 1] ∪ (2, 3). The subsets [0, 1] and (2, 3) of X are both closed and open (they are clopen) as subsets of X. The interval [0, 1] is closed in R while (2, 3) is open in R.

(3) K = \{1, \frac{1}{2}, \frac{1}{3}, …\} is not closed in R but it is closed in R_{K} (8.10).

(4) R_{ℓ} is (8.10) the set of real numbers equipped with the topology generated by the basis sets [a, b) of right half-open intervals. All sets that are (open) closed in the standard topology R are also (open) closed in the finer topology R_{ℓ}. Sets of the form (−∞, a) = ⋃_{x<a}[x, a) = R−[a, ∞), [a, b) = (−∞, b) ∩ [a, ∞), and [a, ∞) = ⋃_{a<x}[a, x) = R − (−∞, a) are both open and closed. Sets of the form (−∞, b] or [a, b] are closed (since they are closed in the standard topology) and not open since they are not unions of basis sets. Sets of the form (a, ∞) are open (since they are open in standard topology) and not closed. Sets of the form (a, b] are neither open nor closed.

(5) Let X be a well-ordered set (§6). In the order topology (§9), sets of the form (a, ∞) = [a^{+}, ∞) = X − (−∞, a], (−∞, b] = (∞, b^{+}) = X − (b, ∞) and (a, b] = (a, ∞) ∩ (−∞, b] are closed and open. (Here, b^{+} denotes b if b is the largest element and the immediate successor of b if b is not the largest element.)

12.3. Closure and interior. We consider the largest open set contained in A and the smallest closed set containing A.

Definition 12.4. The interior of A is the union of all open sets contained in A:

Int A = ⋃{U ⊂ A | U open} = A^{∘}

The closure of A is the intersection of all closed sets containing A:

Cl A = ⋂\{C ⊃ A | C closed\} = \overline{A}

Note [8, Ex 17.6, 17.7, 17.8, 17.9] [3, p 9] [6, 7.1.7–9] that • X − \overline{A} = (X − A)^{∘} and X − A^{∘} = \overline{X − A}

• A^{∘} ⊂ A ⊂ \overline{A}

• A^{∘} is open, it is the largest open set contained in A, and A is open iff A^{∘} = A

• \overline{A} is closed, it is the smallest closed set containing A, and A is closed iff A = \overline{A}

• (A ∩ B)^{∘} = A^{∘} ∩ B^{∘}, \overline{A ∪ B} = \overline{A} ∪ \overline{B}, (A ∪ B)^{∘} ⊃ A^{∘} ∩ B^{∘}, \overline{A ∩ B} ⊂ \overline{A} ∩ \overline{B},

• A ⊂ B ⇒ \{ \begin{matrix}Int A ⊂ Int B \\ Cl A ⊂ Cl B \end{matrix}

The first point, for instance, is the computation

X − \overline{A} = X − ⋂\{C ⊃ A | C closed\} = ⋃\{X − C ⊂ X − A | X − C open\}

= ⋃{U ⊂ X − A | U open} = (X − A)^{∘}

or one can simply say that X − \overline{A} is the largest open subset of X − A since \overline{A} is the smallest closed superset of A.

Definition 12.5. A neighborhood of the point x ∈ X is an open set containing x. A neighborhood of the set A ⊂ X is an open set containing A.

Proposition 12.6. Let A ⊂ X and let x be a point in X. Then

x ∈ A^{∘} ⟺ There exists a neighborhood U of x such that U ⊂ A
x ∈ \overline{A} ⟺ U ∩ A ≠ ∅ for all neighborhoods U of x

Proof. The first assertion is clear as A^{∘} is the union of the open sets contained in A. The second assertion is just a reformulation of the first one,

x ∉ \overline{A} ⟺ x ∈ X − \overline{A} ⟺ x ∈ (X − A)^{∘} ⟺ x has a neighborhood disjoint from A,

since X − \overline{A} = (X − A)^{∘}. □

Proposition 12.7 (Closure with respect to subspace). Let A ⊂ Y ⊂ X. Then (1) A is closed in Y ⟺ A = Y ∩ C for some closed set C in X

(2) If Y is closed then: A is closed in Y ⟺ A is closed in X

(3) Cl_{Y} (A) = Y ∩ \overline{A}.

Proof. (1) Let A be a subset of Y. Then:

A is closed in Y ⟺ Y − A is open in Y
⟺^{11.3.(1)} Y − A = Y ∩ U for some open U ⊂ X
⟺ A = Y ∩ (X − U) for some open U ⊂ X
⟺ A = Y ∩ C for some closed C ⊂ X

(2) If A ⊂ Y and Y is closed then

A is closed in Y ⟺ A = Y ∩ C for some closed C ⊂ X

⟺ A is closed in X

(3) The set Cl_{Y} (A) is the intersection of all relatively closed sets containing A. The relatively closed sets are the sets of the form Y ∩ C where C is closed in X. This means that

Cl_{Y} (A) = ⋂\{Y ∩ C | C ⊃ A, C closed\} = Y ∩ ⋂\{C | C ⊃ A, C closed\} = Y ∩ \overline{A}

by a direct computation. □

If A ⊂ Y ⊂ X, Y ∩Int(A) ⊂ Int_{Y} (A) for Y ∩Int(A) is a relatively open set contained in A. These two sets are equal if Y is open but they are distinct in general (consider A = [0, 1) ⊂ [0, 1] = Y).

Definition 12.8. A subset A ⊂ X is said to be dense if \overline{A} = X, or, equivalently, if every open subset of X contains a point of A.

Proposition 12.9. Let A be a dense and U an open subset of X. Then \overline{A ∩ U} = \overline{U}.

Proof. The inclusion \overline{A ∩ U} ⊂ \overline{U} is general. For the other inclusion, consider a point x ∈ \overline{U}. Let V be any neighborhood of x. Then V ∩ (A ∩ U) = (V ∩ U) ∩ A is not empty since V ∩ U is a neighborhood of x and A is dense. But this says that x is in the closure of A ∩ U. □

12.10. Limit points and isolated points. Let X be a topological space and A a subset of X.

Definition 12.11 (Limit points, isolated points). A point x ∈ X is a limit point of A if U ∩ (A − {x}) ≠ ∅ for all neighborhoods U of x. The set of limit points^{1} of A is denoted A′. A point a ∈ A is an isolated point if a has a neighborhood that intersects A only in {a}.

Equivalently, x ∈ X is a limit point of A iff x ∈ \overline{A − \{x\}}, and a ∈ A is an isolated point of A iff {a} is open in A. These two concepts are almost each others opposite:

x is not a limit point of A ⟺ x has a neighborhood U such that U ∩ (A ∪ {x}) = {x}

⟺ x is an isolated point of A ∪ {x}

Proposition 12.12. Let A be a subset of X and A′ the set of limit points of A. Then A ∪ A′ = \overline{A} and A ∩ A′ = {a ∈ A | a is not an isolated point of A} so that

A ⊃ A′ ⟺ A is closed
A ⊂ A′ ⟺ A has no isolated points
A ∩ A′ = ∅ ⟺ A is discrete
A′ = ∅ ⟺ A is closed and discrete

If B ⊂ A then B′ ⊂ A′.

1The set of limit points of A is sometimes denotes A^{d} and called the derived set of A

Proof. It is clear that all limit points of A are in \overline{A}, so that A ∪ A′ ⊂ \overline{A}, and that all points in the closure of A that are not in A are limit points,

\overline{A} − A = \{x ∈ X − A | all neighborhoods of x meet A\} ⊂ A′ ⊂ \overline{A},

so that \overline{A} = A ∪ (\overline{A} − A) ⊂ A ∪ A′. This shows that A ∪ A′ = \overline{A}. From the above discussion we have that the set of limit points in A, A ∩ A′ = A − (A − A′), is the set of non-isolated points of A. If A ∩ A′ = ∅ then all all points of A are isolated so that the subspace A has the discrete topology. We have A′ = ∅ ⟺ A ⊃ A′, A ∩ A′ = ∅ ⟺ A is closed and discrete. □

Example 12.13. (1) \overline{Q} = R = \overline{R − Q} (R−Q contains (Q−\{0\})\sqrt{2}). Cl Int Q = ∅, Int Cl Q = R.

(2) Let C_{r} ⊂ R^{2} be the circle with center (0, r) and radius r > 0. Then

\overline{⋃_{n∈Z_{+}} C_{1/n}} = ⋃_{n∈Z_{+}} C_{1/n}, \overline{⋃_{n∈Z_{+}} C_{n}} = (R × \{0\}) ∪ ⋃_{n∈Z_{+}} C_{n}

The first set of decreasing circles (known as the Hawaiian Earring [8, Exmp 1 p 436]) is closed because it is also the intersection of a collection (which collection?) of closed sets.

(3) The set of limit points of K = \{\frac{1}{n} | n ∈ Z_{+}\} is \{0\} in R and R_{ℓ} and ∅ in R_{K}.

(4) Let X be a linearly ordered space (9.1). Closed intervals [a, b] are closed because their complements X − [a, b] = (−∞, a) ∪ (b, ∞) are open. Therefore the closure of an interval of the form [a, b) is either [a, b) or [a, b]. If b has an immediate predecessor b^{−} then [a, b) = [a, b^{−}] is closed so that \overline{[a, b)} = [a, b). Otherwise, \overline{[a, b)} = [a, b] because any neighborhood of b contains an interval of the form (c, b] for some c < b and [a, b) ∩ (c, b], which equals [a, b] or (c, b], is not empty.

12.14. Convergence, the Hausdorff property, and the T_{1}-axiom. Let x_{n}, n ∈ Z_{+}, be a sequence of points in X, ie a map Z_{+} → X.

Definition 12.15. The sequence x_{n} converges to the point x ∈ X if for any neighborhood U of x there exists some N such that x_{n} ∈ U for all n > N.

If a sequence converges in some topology on X it also converges in any coarser topology but not necessarily in a finer topology.

Example 12.16. In the Sierpinski space X = {0, 1} (8.2.(3)), the sequence 0, 0, ⋯ converges to 0 and to 1. In R with the finite complement topology (8.2.(4)), the sequence 1, 2, 3, ⋯ converges to any point. In R_{K}, the sequence \frac{1}{n} does not converge; in R and R_{ℓ} it converges to 0 (and to no other point). The sequence −\frac{1}{n} converges to 0 in R and R_{K}, but does not converge in R_{ℓ}. In the ordered space [1, ω] = Z_{+} ⨿ {ω} (6.5.(6)), the sequence n converges to ω (and to no other point). Can you find a sequence in the ordered space [0, Ω] (6.8) that converges to Ω?

Definition 12.17 (Separation Axioms). A topological space is T_{1}-space if points are closed: For any two distinct points x_{1} ≠ x_{2} in X there exists an open set U such that x_{1} ∈ U and x_{2} ∉ U.

A topological space X is a T_{2}-space (or a Hausdorff space) if there are enough open sets to separate points: For any two distinct points x_{1} ≠ x_{2} in X there exist disjoint open sets, U_{1} and U_{2}, such that x_{1} ∈ U_{1} and x_{2} ∈ U_{2}.

All Hausdorff spaces are T_{1}. X is T_{1} iff all finite subsets are closed. Cofinite topologies are T_{1} by construction and not T_{2} when the space has infinitely many points. Particular point topologies are not T_{1} (on spaces with more than one point).

All linearly ordered [Ex 17.10] or metric spaces are Hausdorff; in particular, R is Hausdorff. R_{ℓ} and R_{K} are Hausdorff because the topologies are finer than the standard Hausdorff topology R.

Theorem 12.18. A sequence in a Hausdorff space can not converge to two distinct points.

A property of a topological space is said to be (weakly) hereditary if any (closed) subspace of a space with the property also has the property. Hausdorffness is hereditary and also passes to product spaces.

Theorem 12.19 (Hereditary properties of Hausdorff spaces). [Ex 17.11, 17.12] Any subset of a Hausdorff space is Hausdorff. Any product of Hausdorff spaces is Hausdorff.

Theorem 12.20. Suppose that X is T_{1}. Let A be a subset of and x a point in X. Then

x is a limit point ⟺ All neighborhoods of x intersect A in infinitely many points

Proof. ⟸: If all neighborhoods of x intersect A in infinitely many points, then, clearly, they also intersect A in a point that is not x. ⟹: Assume that x is a limit point and let U be a neighborhood of x. Then U contains a point a_{1} of A different from x. Remove this point from U. Since points are closed, U − {a_{1}} is a new neighborhood of x. This new neighborhood of x contains a point a_{2} of A different from x and a_{1}. In this way we recursively find a whole sequence of distinct points in U ∩ A. □

13. Continuous functions

Let f: X → Y be a map between two topological spaces.

Definition 13.1. The map f: X → Y is continuous if all open subsets of Y have open preimages in X: V open in X ⟹ f^{−1}(V) open in Y

If T_{X} is the topology on X and T_{Y} is the topology on Y then

(13.2)  f is continuous ⟺ f^{−1}(T_{Y}) ⊂ T_{X} ⟺ f^{−1}(B_{Y}) ⊂ T_{X} ⟺ f^{−1}(S_{Y}) ⊂ T_{X}

where B_{Y} is a basis and S_{Y} a subbasis for T_{Y}. The finer the topology in Y and the coarser the topology in X are, the more difficult is it for f: X → Y to be continuous.

Theorem 13.3. Let f: X → Y be a map between two topological spaces. The following are equivalent:

(1) f is continuous

(2) The preimage of any open set in Y is open in X: V open in Y ⟹ f^{−1}(V) open in X (3) The preimage of any closed set in Y is closed in X: C closed in Y ⟹ f^{−1}(C) closed in X (4) f^{−1}(B^{∘}) ⊂ (f^{−1}(B))^{∘} for any B ⊂ Y

(5) \overline{f^{−1}(B)} ⊂ f^{−1}(\overline{B}) for any B ⊂ Y

(6) f(\overline{A}) ⊂ \overline{f(A)} for any A ⊂ X

(7) For any point x ∈ X and any neighborhood V ⊂ Y of f(x) there is a neighborhood U ⊂ X of x such that f(U) ⊂ V.

Proof. It is easy to see that (7),(1),(2), and (3) are equivalent.

(2) ⟹ (4): \begin{matrix}f^{−1}(B^{∘}) ⊂ f^{−1}(B) \\ f^{−1}(B^{∘}) open \end{matrix} \} ⟹ f^{−1}(B^{∘}) ⊂ (f^{−1}(B))^{∘}.

(4) ⟹ (2): Let B be any open set in Y. Then f^{−1}(B) =^{B=B^{∘}} f^{−1}(B^{∘}) ⊂^{(4)} (f^{−1}(B))^{∘} ⊂ f^{−1}(B) so f^{−1}(B) is open since it equals its own interior.

(3) ⟺ (5): Similar to (2) ⟺ (4).

(3) ⟹ (6): \begin{matrix}A ⊂ f^{−1}(\overline{f(A)}) \\ f^{−1}(\overline{f(A)}) is closed \end{matrix} \} ⟹ \overline{A} ⊂ f^{−1}(\overline{f(A)}) ⟺ f(\overline{A}) ⊂ \overline{f(A)}.

(6) ⟹ (5): f(\overline{f^{−1}(B)}) ⊂^{(6)} \overline{f(f^{−1}(B))} ⊂^{f(f^{−1}(B))⊂B} \overline{B}  □

Example 13.4. The function f(x) = −x is continuous R → R, but not continuous R_{K} → R_{K} (K is closed in R_{K} but f^{−1}(K) is not closed (12.16)) and not continuous R_{ℓ} → R_{ℓ} ([0, ∞) is open in R_{ℓ} but f^{−1}([0, ∞)) = (−∞, 0] is not open (12.2.(4))). Since [a, b) is closed and open in R_{ℓ} (12.2.(4)) the map 1_{[a,b)} : R_{ℓ} → {0, 1}, with value 1 on [a, b) and value 0 outside [a, b), is continuous.

Theorem 13.5. (1) The identity function is continuous.

(2) The composition of two continuous functions is continuous.

(3) Let B be a subspace of Y, A a subspace of X, and f: X → Y a map taking values in B. Then

f: X → Y is continuous ⟹ f|A: A → Y is continuous  (restriction)
f: X → Y is continuous ⟺ B|f: X → B is continuous  (corestriction)

(4) Let f_{j}: X_{j} → Y_{j}, j ∈ J, be an indexed set of maps. Then

∏ f_{j}: ∏ X_{j} → ∏ Y_{j} is continuous ⟺ f_{j}: X_{j} → Y_{j} is continuous for all j ∈ J

(5) Suppose that X = ⋃ U_{α} is a union of open sets. Then

f: X → Y is continuous ⟺ f|U_{α} is continuous for all α

for any map f: X → Y.

(6) (Glueing lemma) Suppose that X = C_{1} ∪ ⋯ ∪ C_{n} is a finite union of closed sets. Then

f: X → Y is continuous ⟺ f|C_{i} is continuous for all i = 1, …, n

for any map f: X → Y.

Proof. This is easy to check. The ⟹-part of (4) uses 13.14 below. □

13.6. Homeomorphisms and embeddings. One of the central problems in topology is to decide if two given spaces are homeomorphic.

Definition 13.7 (Homeomorphism). A bijective continuous map f: X → Y is a homeomorphism if its inverse is continuous.

A bijection f: X → Y is a homeomorphism iff

U is \begin{cases}open \\ closed \end{cases}\} in X ⟺ f(U) is \begin{cases}open \\ closed \end{cases}\} in Y or

f^{−1}(V) is \begin{cases}open \\ closed \end{cases}\} in X ⟺ V is \begin{cases}open \\ closed \end{cases}\} in Y

holds for all U ⊂ X or V ⊂ Y.

We now extend the subspace topology (11.1) to a slightly more general situation.

Definition 13.8 (Subspace topology). Let X be a set, Y a topological space, and f: X → Y an injective map. The subspace topology on X (wrt to the map f) is the collection

f^{−1}(T_{Y}) = {f^{−1}(V) | V ⊂ Y open}

of subsets of X.

The subspace topology is the coarsest topology on X such that f: X → Y is continuous. There are as few open sets in X as possible without destroying continuity of f: X → Y.

Proposition 13.9 (Characterization of the subspace topology). Suppose that X has the subspace topology wrt to the map f: X → Y. Then

(1) X → Y is continuous and,

(2) for any map A → X into X,

A → X is continuous ⟺ A → X ⟶^{f} Y is continuous

The subspace topology is the only topology on X with these properties.

Proof. This is because

A ⟶^{g} X is continuous ⟺^{(13.2)} g^{−1}(T_{X}) ⊂ T_{A} ⟺ g^{−1}(f^{−1}T_{Y}) ⊂ T_{A} ⟺

(fg)^{−1}T_{Y}) ⊂ T_{A} ⟺ A ⟶^{g} X ⟶^{f} Y is continuous

by definition of the subspace topology. The identity map of X is a homeomorphism whenever X is equipped with a topology with these two properties. □

Definition 13.10 (Embedding). An injective continuous map f: X → Y is an embedding if X has the subspace topology (ie X has the smallest topology such that f is continuous, or T_{X} = f^{−1}(T_{Y})).

This means that f is an embedding if and only if for all U ⊂ X:

(13.3) U is \begin{cases}open \\ closed \end{cases}\} in X ⟺ U = f^{−1}(V) for some \begin{cases}open \\ closed \end{cases}\} V ⊂ Y

⟺ f(U) = f(X) ∩ V for some \begin{cases}open \\ closed \end{cases}\} V ⊂ Y ⟺ f(U) is \begin{cases}open \\ closed \end{cases}\} in f(X)

Alternatively, the injective map f: X → Y is an embedding if and only if the bijective map f(X)|f: X → f(X) is a homeomorphism. An embedding is a homeomorphism followed by an inclusion. The inclusion A ↪ X of a subspace is an embedding. Any open (closed) continuous injective map is an embedding.

Example 13.11. (1) The map f(x) = 3x + 1 is a homeomorphism R → R.

(2) The identity map R_{ℓ} → R is bijective and continuous but not a homeomorphism.

(3) The map [0, 1) → S^{1} : t ↦ (cos(2πt), sin(2πt)) is continuous and bijective but not a homemorphism. The image of the open set [0, \frac{1}{2}) is not open in S^{1}.

(4) Find an example of an injective continuous map R → R^{2} that is not an embedding.

(5) The obvious bijection [0, 1) ∪ {2} → [0, 1] is continuous but not a homeomorphism (the domain has an isolated point, the codomain has no isolated points). There does not exist any continuous surjection in the other direction.

(6) The spaces [1 × 1, 2 × 1] ⊂ Z_{+} × Z_{+} and \overline{K} = \overline{\{\frac{1}{n} | n ∈ Z_{+}\}} ⊂ R are homeomorphic.

(7) The map R → R × R: t ↦ (t, t) is an embedding. For any continuous map f: X → Y, the map X → X × Y : x ↦ (x, f(x)) is an embedding; see (25.4) for a generalization.

(8) R^{n} embeds into S^{n} via stereographic projection.

(9) R^{2} embeds in R^{3}. Does R^{3} imbed in R^{2}? (See notes on algebraic topology for the answer.)

(10) Are the spaces ⋃ C_{n} and ⋃ C_{1/n} of 12.13.(2) homeomorphic?

(11) A knot is in embedding of S^{1} in R^{3} (or S^{3}). Two knots, K_{0} : S^{1} ↪ R^{3} and K_{1} : S^{1} ↪ R^{3}, are equivalent if there exists a homeomorphism h of R^{3} such that h(K_{0}) = K_{1}. The fundamental problem of knot theory [1] is to classify knots up to equivalence.

Lemma 13.12. If f: X → Y is a homeomorphism (embedding) then the corestriction of the restriction f(A)|f|A: A → f(A) (B|f|A: A → B) is a homeomorphism (embedding) for any subset A of X (and any subset B of Y containing f(A)). If the maps f_{j}: X_{j} → Y_{j} are homeomorphisms (embeddings) then the product map ∏ f_{j}: ∏ X_{j} → Y_{j} is a homeomorphism (embedding).

Proof. In case of homeomorphisms there is a continuous inverse in both cases. In case of embeddings, use that an embedding is a homeomorphism followed by an inclusion map. □

13.13. Maps into products. There is an easy test for when a map into a product space is continuous.

Theorem 13.14 (Characterization of the product topology). Give ∏ Y_{j} the product topology. Then

(1) the projections π_{j}: ∏ Y_{j} → Y_{j} are continuous, and,

(2) for any map f: X → ∏_{j∈J} Y_{j} into the product space we have

X ⟶^{f} ∏_{j∈J} Y_{j} is continuous ⟺ ∀j ∈ J: X ⟶^{f} ∏_{j∈J} Y_{j} ⟶^{π_{j}} Y_{j} is continuous

The product topology is the only topology on the product set with these two properties.

Proof. Let T_{X} be the topology on X and T_{j} the topology on Y_{j}. Then S∏ = ⋃_{j∈J} π_{j}^{−1}(T_{j}) is a subbasis for the product topology on ∏_{j∈J} Y_{j} (10.1). Therefore

f: X → ∏_{j∈J} Y_{j} is continuous ⟺^{(13.2)} f^{−1}( ⋃_{j∈J} π_{j}^{−1}(T_{j})) ⊂ T_{X}

⟺ ⋃_{j∈J} f^{−1}(π_{j}^{−1}(T_{j})) ⊂ T_{X}

⟺ ∀j ∈ J: (π_{j} ∘ f)^{−1}(T_{j}) ⊂ T_{X} ⟺ ∀j ∈ J: π_{j} ∘ f is continuous by definition of continuity (13.2).

We now show that the product topology is the unique topology with these properties. Take two copies of the product set ∏_{j∈J} X_{j}. Equip one copy with the product topology and the other copy with some topology that has the two properties of the theorem. Then the identity map between these two copies is a homeomorphism. □

The reason for the great similarity between 13.14 and 13.9 is that in both cases we use an initial topology.

Example 13.15. Suppose that J and K are sets and that (X_{j})_{j∈J} and (Y_{k})_{k∈k} are indexed families of topological spaces. Given a map g: J → K between index sets and an indexed family of continuous maps (f_{j}: Y_{g(j)} → X_{j})_{j∈J}. Then there is a unique map between product spaces such that

\xymatrix{∏_{k∈K} Y_{k} \ar[r] \ar[d]_{π_{g(j)}} & ∏_{j∈J} X_{j} \ar[d]^{π_{j}} \\ Y_{g(j)} \ar[r]_{f_{j}} & X_{j}}

commutes and this map of product spaces is continuous by 13.14.

Theorem 13.16. Let (X_{j})_{j∈J} be an indexed family of topological spaces with subspaces A_{j} ⊂ X_{j}. Then ∏ A_{j} is a subspace of ∏ X_{j}.

(1) \overline{∏ A_{j}} = ∏ \overline{A_{j}}.

(2) ( ∏ A_{j})^{∘} ⊂ ∏ A_{j}^{∘} and equality holds if A_{j} = X_{j} for all but finitely many j ∈ J.

Proof. (1) Let (x_{j}) be a point of ∏ X_{j}. Since B∏ = ⋃_{j∈J} π_{j}^{−1}(U_{j}) is a subbasis for the product topology we have:

(x_{j}) ∈ \overline{∏ A_{j}} ⟺ ∀k ∈ J: π_{k}^{−1}(U_{k}) ∩ ∏ A_{j} ≠ ∅ for all neighborhoods U_{k} of x_{k}
⟺ ∀k ∈ J: U_{k} ∩ A_{k} ≠ ∅ for all neighborhoods U_{k} of x_{k}
⟺ ∀k ∈ J: x_{k} ∈ \overline{A_{k}}
⟺ (x_{j}) ∈ ∏ \overline{A_{j}}

(2) ( ∏ A_{j})^{∘} ⊂ ∏ A_{j}^{∘} because π_{j} is an open map (14.3) so that π_{j} (( ∏ A_{j})^{∘}) ⊂ A_{j}^{∘} for all j ∈ J. If

A_{j} = X_{j} for all but finitely many j ∈ J then ∏ A_{j}^{∘} ⊂ ( ∏ A_{j})^{∘} because ∏ A_{j}^{∘} is open and contained in ∏ A_{j}. □

It follows that a product of closed sets is closed. (Whereas a product of open sets need not be open in the product topology.)

13.17. Maps out of coproducts.

Theorem 13.18. Let f: ∐_{j∈J} X_{j} → Y be a map out of a coproduct space. Then

f: ∐_{j∈J} X_{j} → Y is continuous ⟺ f ∘ ι_{j}: X_{j} → Y is continuous for all j ∈ J

where ι_{j}: X_{j} → ∐_{j∈J} X_{j} is the inclusion map.

Let f_{j}: X_{j} → Y_{j}, j ∈ J, be an indexed set of maps. Then

∐ f_{j}: ∐ X_{j} → ∐ Y_{j} is continuous ⟺ f_{j}: X_{j} → Y_{j} is continuous for all j ∈ J

14. The quotient topology

In this section we will look at the quotient space construction. But first we consider open and closed maps.

14.1. Open and closed maps. [2, I.§5] Let X and Y be topological spaces and f: X → Y a map.

Definition 14.1. The map f: X → Y is \begin{cases}open \\ closed \end{cases}\} if for all U ⊂ X we have:

U \begin{cases}open \\ closed \end{cases}\} in X ⟹ f(U) \begin{cases}open \\ closed \end{cases}\} in Y

The restriction (14.13) of an open (or closed) map to an arbitrary subspace need not be open (closed). However,

Proposition 14.2. [8, Ex 22.5] The restriction of an open (or closed) map f: X → Y to an open (closed) subspace A ⊂ X is an open (closed) map f(A)|f|A: A → f(A) or f|A: A → Y .

Proof. Suppose that f: X → Y is an open map and A ⊂ X an open subset. Let U be an open subset of A. The implications

U is open in A  (A is open in X)
⟹ U open in X  (f is open)
⟹ f(U) open in Y
⟹ f(U) open in f(A)

show that f|A: A → f(A) is open. □

Proposition 14.3 (Projections are open). The projection map π_{j}: ∏ X_{j} → X_{j} is open.

Proof. The map π_{j} takes the basis B∏ (10.1) for the product topology into the topology on X_{j}. □

Lemma 14.4 (Characterization of open or closed continuous maps). Let f: X → Y be a continuous map.

(1) f is open if and only if f^{−1}(Int B) = Int f^{−1}(B) for all B ⊂ Y.

(2) f is closed if and only if f(\overline{A}) = \overline{f(A)} for all A ⊂ X

Proof. See Solution June 04 (Problem 1) and Solution Jan 05 (Problem 1). □

A bijective continuous map that is open or closed is a homeomorphism.

Example 14.5. (1) The projection π_{1} : X × Y → X is continuous and open (14.3). It is closed if Y is compact. (Use 18.10 to see this.) The projection map π_{1} : R × R → R is not closed for H = {(x, y) ∈ R × R | xy = 1} is closed but π_{1}(H) = R − {0} is not closed.

(2) The map f: [−1, 2] → [0, 1] given by

f(x) = \begin{cases}0 & −1 ≤ x ≤ 0 \\ x & 0 ≤ x ≤ 1 \\ 1 & 1 ≤ x ≤ 2 \end{cases}

is continuous and closed (18.8.(1)). It is not open for f([−1, −1/2)) = {0} is not open.

14.2. Quotient topologies and quotient maps. Quotient maps are continuous surjective maps that generalize both continuous, open surjective maps and and continuous, closed surjective maps.

Definition 14.6 (Quotient topology). Let X be a topological space, Y a set, and p: X → Y a surjective map. The quotient topology on Y is the collection

{V ⊂ Y | p^{−1}(V) is open in X}

of subsets of Y.

The quotient topology^{2} is the finest topology on Y such that p: X → Y continuous. There are as many open sets in Y as possible without destroying continiuty of f: X → Y. Quotient topology and subspace topology (13.8) are ‘dual’ concepts .

Lemma 14.7 (Characterization of the quotient topology). Suppose that Y has the quotient topology with respect to the map p: X → Y. Then

(1) p: X → Y is continuous, and,

(2) for any map g: Y → Z out of Y

Y ⟶^{g} Z is continuous ⟺ X ⟶^{p} Y ⟶^{g} Z is continuous

The quotient topology is the only topology on Y with these properties.

Proof. This is because

Y ⟶^{g} Z is continuous ⟺ g^{−1}(T_{Z}) ⊂ T_{Y} ⟺ p^{−1}g^{−1}(T_{Z}) ⊂ T_{X}

⟺ (pg)^{−1}(T_{Z}) ⊂ T_{X} ⟺ X ⟶^{p} Y ⟶^{g} Z is continuous

by definition of the quotient topology.

If we give Y some topology with these two properties then the identity map between the two topologies is a homeomorphism. □

Definition 14.8 (Quotient map). A surjective continuous map p: X → Y is a quotient map if Y has the quotient topology.

Quotient maps and embeddings (13.10) are dual concepts.

A surjective map p: X → Y is a quotient map when the quotient topology on Y, T_{Y} = {V ⊂ Y | p^{−1}(V) is open in X}, is the finest topology on Y such that p is continuous. This means that p is a quotient map if and only if

p^{−1}(V) is open in X ⟺ V is open in Y

holds for all V ⊂ Y.

Sets of the form p^{−1}(V) = ⋃_{y∈V} p^{−1}(y), consisting of unions of fibres, are called saturated sets. The saturation of A ⊂ X is the union f^{−1}f(A) = ⋃_{y∈f(A)} f^{−1}(y) of all fibres that meet A.

Proposition 14.9. For a surjective map p: X → Y the following are equivalent:

(1) p: X → Y is a quotient map

(2) For all V ⊂ Y we have: p^{−1}(V) is \begin{cases}open \\ closed \end{cases}\} in X ⟺ V is \begin{cases}open \\ closed \end{cases}\} in Y.

(3) p: X → Y is continuous and maps saturated \begin{cases}open \\ closed \end{cases}\} sets to \begin{cases}open \\ closed \end{cases}\} open sets

Proof. Condition (1) and condition (2) with the word “open” are clearly equivalent. Suppose now that: p^{−1}(V) is open ⟺ V is open. Then we get

p^{−1}(C) is closed ⟺ X − p^{−1}(C) is open ⟺ p^{−1}(Y − C) is open ⟺ Y − C is open ⟺ C is closed

for all C ⊂ Y. This shows that the two conditions of (2) are equivalent. The content of (3) is just a reformulation of (2). □

Thus a quotient map p: X → Y induces a bijective correspondence

[figure: \begin{cases}Saturated open (closed) sets \\ in X \end{cases}\}; U→p(U); p^{−1}(V)←V; \begin{cases}Open (closed) sets \\ in Y \end{cases}\}]

Just as for open (closed) continuous bijective maps we evidently have:

Corollary 14.10. A bijective continuous map is a quotient map if and only if it is a homeomorphism.

Quotient maps behave nicely with respect to composition:

2The quotient topology is sometimes called the final topology [2, I.§2.4] with respect to the map f

Corollary 14.11. Let \xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z} be continuous maps. Then

f and g are quotient ⟹ g ∘ f is quotient ⟹ g is quotient

Proof. The first assertion is a tautology: Assume that f and g are quotient. Then

(g ∘ f)^{−1}(V) open in X ⟺ f^{−1}g^{−1}(V) open in X ⟹^{f is quotient} g^{−1}(V) open in Y

⟹^{g is quotient} V open in Z

for any set V ⊂ Y. Next, suppose that X ⟶^{f} Y ⟶^{g} Z is a quotient map. Then the last map g is surjective and

g^{−1}(V) is open in Y ⟹^{f continuous} (g ∘ f)^{−1}(V) is open in X ⟹^{g ∘ f quotient} V is open in Z

for any set V ⊂ Z. □

Corollary 14.12. (1) Any \begin{cases}open \\ closed \end{cases}\} continuous surjective map is a quotient map.

(2) A quotient map f: X → Y is \begin{cases}open \\ closed \end{cases}\} if and only if all \begin{cases}open \\ closed \end{cases}\} sets A ⊂ X have

\begin{cases}open \\ closed \end{cases}\} saturations f^{−1}f(A).

Proof. Let f: X → Y be an open map. Then

f^{−1}(V) is open ⟹^{f open} ff^{−1}(V) is open ⟺ V is open ⟹^{f is continuous} f^{−1}(V) is open

for all V ⊂ Y. This shows that f is quotient. Suppose next that f: X → Y is quotient. Then

f is open ⟺^{(14.1)} f(A) is open in Y for all open sets A ⊂ X
⟺^{(14.8)} f^{−1}f(A) is open in X for all open sets A ⊂ X

which is the claim. □

There are quotient maps that are neither open nor closed [8, Ex 22.3] [2, Ex 10 p 135] and there are (non-identity) quotient maps that are both open and closed (14.15) [2, Ex 3 p 128].

Example 14.13. (1) The projection map π_{1} : R × R → R is (14.5.(1)) open, continuous, and surjective so it is a quotient map. The restriction π_{1}|H ∪ {(0, 0)} is continuous and surjective, even bijective, but it is not a quotient map (14.9) for it is not a homeomorphism: {(0, 0)} is open and saturated in H ∪ {(0, 0)} but π_{1}({(0, 0)} = {0} is not open.

Thus the restriction of a quotient map need not be a quotient map in general. On the positive side we have

Proposition 14.14. The restriction of a quotient map p: X → Y to an open (or closed) saturated subspace A ⊂ X is a quotient map p|A: A → p(A).

Proof. (Similar to the proof of 14.2.) Let p: X → Y be a quotient map and B ⊂ Y an open set. (The case where B is closed is similar.) The claim is that p|p^{−1}(B): p^{−1}(B) → B is quotient. For any U ⊂ B the implications

p^{−1}(U) open in p^{−1}(B)  (p^{−1}(B) is open)
⟹ p^{−1}(U) open in X  (p is quotient)

⟹ U is open in Y ⟹ U is open in B

show that p|p^{−1}(B): p^{−1}(B) → B is quotient. □

A typical situation is when R is an equivalence relation on the space X and X → X/R is the map that takes a point to its equivalence class. We call X/R with the quotient topology for the quotient space of the equivalence relation R. A set of equivalence classes is an open subset of X/R if and only if the union of equivalence classes is an open subset of X: V ⊂ X/R is open ⟺ ⋃_{[x]∈V} [x] ⊂ X is open. We shall often say that X/R is the space obtained by identifying equivalent points of X.

Example 14.15 (Orbit spaces). (1) The real projective n-space RP^{n} is the quotient space of S^{n} by the equivalence relation with equivalence classes {±x}, x ∈ S^{n}. The quotient map p: S^{n} → RP^{n} is both open and closed since (14.12) the saturation ±U of an open (closed) set U ⊂ S^{n} is open (closed) because x → −x is a homeomorphism. Elements of RP^{n} can be thought of as lines through the origin of R^{n+1}. A set of lines is open if the set of intersection points with the unit sphere is open.

(2) More generally, let G × X → X be the action of a discrete group G on a space X. Give the orbit space G\X the quotient topology and let p_{G}: X → G\X be the quotient map. The points in the orbit space are orbits of points in X and the open subsets are orbits of open subsets of X. The saturation of any subset A of X is the orbit GA = ⋃_{g∈G} gA of A. If A is open, GA is open; if A is closed and G is finite, GA is closed. Thus the quotient map p_{G} is always open (14.12), and if G is finite, it is also closed.

A continuous map f: X → Y respects the equivalence relation R if equivalent points have identical images, that is if x_{1}Rx_{2} ⟹ f(x_{1}) = f(x_{2}). The quotient map X → X/R respects the equivalence relation R and it is the universal example of such a map.

Theorem 14.16. Let f: X → Y be a continuous map. Then (1) The map p: X → X/R respects the equivalence relation R.

(2) If the continuous map f: X → Y respects R then there exists a unique continuous map \overline{f}: X → Y such that

\xymatrix{X \ar[rr]^{f} \ar[dr]_{p} &  & Y \\  & X/R \ar@{-->}[ur]_{\overline{f}}}

commutes. (We say that f factors uniquely through X/R.) Conversely, if f factors through X/R then f respects R.

(3) If \overline{f} exists then: \overline{f} is quotient ⟺ f is quotient

Proof. If \overline{f} exists then clearly f respects R. Conversely, if f respects R then we can define \overline{f}[x] = f(x) and this map is continuous by 14.7 and it is the only possibility. The rest follows from 14.11. □

The theorem says that there is a bijective correspondence

[figure: \begin{cases}Continuous maps X → Y \\ that respect R \end{cases} \}; f→\overline{f}; g∘p←g; \begin{cases}Continuous maps \\ X/R → Y \end{cases} \}]

taking quotient maps to quotient maps.

Example 14.17. (1) Let f: X → Y be any surjective continuous map. Consider the equivalence relation corresponding to the partition X = ⋃_{y∈Y} f^{−1}(y) of X into fibres fibres f^{−1}(y), y ∈ Y. Let X/f denote the quotient space. Thus X/f is the set of fibres equipped with the quotient topology. By constuction, the map f respects this equivalence relation so there is a unique continuous map \overline{f} such that the diagram

\xymatrix{X \ar[dr] \ar[rr]^{f} &  & Y \\  & X/f \ar[ur]_{\overline{f}}}

commutes. Note that \overline{f} is bijective. The bijective continuous map \overline{f}: X/f → Y is a homeomorphism if and only if f is quotient (14.10, 14.16). In particular, all quotient maps have (up to homeomorphism) the form X → X/R for some equivalence relation R on X.

(2) Let f: X → Y be any surjective continuous map. The induced map \overline{f}: X/f → Y is a continuous bijection but in general not a homeomorphism. Instead, the topology on the quotient space X/f is finer than the topology on Y. This can sometimes be used to show that X/f is Hausdorff if Y is Hausdorff.

(3) Let f: X → Y be any continuous map. Then f has a canonical decomposition

X ⟶^{p} X/f ⟶^{\overline{f}} f(X) ⟶^{ι} Y

where p is a quotient map, \overline{f} is a continuous bijection, and ι is an inclusion map.

(4) Let X be a topological space and A_{1}, A_{2}, …, A_{k} a finite collection of closed subsets. Consider the equivalence relation where the equivalence classes are the sets A_{1}, A_{2}, …, A_{k} together with the sets {x} for x ∉ A_{1}∪A_{2}∪⋯∪A_{k}. The quotient space X/(A_{1}, …, A_{k}) is obtained from X by identifying each of the sets A_{i} to the point p(A_{i}). The quotient map p: X → X/(A_{1}, …, A_{k}) is closed because (14.12) closed sets A ⊂ X have closed saturations A∪⋃_{A_{i}∩A≠∅} A_{i}. A continuous map f: X → Y factors through the quotient space X/(A_{1}, …, A_{k}) if and only if it sends each of the sets A_{i} ⊂ X to a point in Y (14.16). The restriction p|X − (A_{1} ∪ ⋯ ∪ A_{k}): X − (A_{1} ∪ ⋯ ∪ A_{k}) → X/(A_{1}, …, A_{k}) − {p(A_{1}), …, p(A_{k})} to the complement of A_{1} ∪ ⋯ ∪ A_{k} is a homeomorphism (14.2). In case of just one closed subspace A ⊂ X, the quotient space is denoted X/A.

(5) The standard map f: [0, 1] → S^{1} that takes t ∈ [0, 1] to (cos(2πt), sin(2πt)) is quotient because it is continuous and closed. (If you can’t see this now, we will prove it later (18.7.(1)).) The induced map [0, 1]/{0, 1} → S^{1} is a homeomorphism. More generally, the standard map D^{n}/S^{n−1} → S^{n} is a homeomorphism where D^{n} ⊂ R^{n}, the unit disc, is the set of vectors of length ≤ 1.

(6) Let R be the equivalence relation “zero or not zero” on R. The quotient space R/R is homeomorphic to Sierpinski space {0, 1}.

(7) There is an obvious continuous surjective map f: Z_{+} × S^{1} → ⋃ C_{n} (12.13.(2)) that takes Z_{+} × {1} to the point common to all the circles. This map is continuous because its restriction to each of the open sets {n} × S^{1} is continuous (13.5.(5). However, f is not a quotient map (14.9) for the image of the closed saturated set consisting of the points n × (cos(\frac{π}{2} ), sin(\frac{π}{2} )) is not closed as it does not contain all its limit points. The induced bijective continuous map \overline{f}: Z_{+} × S^{1}/Z_{+} × \{1\} → ⋃ C_{n} is therefore not a homeomorphism. There is an obvious continuous surjective map g: Z_{+} × S^{1} → ⋃ C_{1/n} (12.13.(2)) that takes Z_{+} × {1} to the point common to all the circles. This map is continuous because its restriction to each of the open sets {n} × S^{1} is continuous (13.5.(5)). However, g is not a quotient map for the image of the closed saturated set Z_{+} × {−1} is not closed as it does not contain all its limit points. The induced bijective continuous map \overline{g}: Z_{+} × S^{1}/Z_{+} × \{1\} → ⋃ C_{1/n} is therefore not a homeomorphism either. (Actually (15.10.(7)), the quotient space Z_{+} × S^{1}/Z_{+} × {1}, known as the countable wedge of circles ⋁_{n∈Z_{+}} S^{1} [8, Lemma 71.4], is not homeomorphic to any subspace of the plane.)

(8) [8, p 451] Let P_{4g} be a regular 4g-gon and with edges labeled a_{1}, b_{1}, a_{1}, b_{1}, …, a_{g}, b_{g}, a_{g}, b_{g} in counter-clockwise direction. The closed orientable surface M_{g} of genus g ≥ 1 is (homeomorphic to) the quotient space P_{4g}/R where R is the equivalence relation that makes the identifications a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1} on the perimeter and no identifications in the interior of the polygon. See [8, p 452] for the case g = 2. Are any of these surfaces homeomorphic to each other? [8, Thm 77.5]

(9) [8, p 452] Let P_{2g} be a regular 2g-gon with edges labeled a_{1}, a_{1}, …, a_{g}, a_{g} in counter-clockwise direction. The closed non-orientable surface N_{g} of genus g ≥ 1 is the quotient space P_{2g}/R where R is the equivalence relation that makes the identifications a_{1}^{2} ⋯ a_{g}^{2} on the perimeter and no identifications in the interior of the polygon. For g = 1 we get the projective plane RP^{2} and for g = 2 we get the Klein Bottle [8, Ex 74.3].

Example 14.18. (The adjunction space) [8, Ex 35.8] [5, p 93] [9, Chp 1, Exercise B p 56] Consider

the set-up \xymatrix{X & A \ar@{^{(}->}[l]_{i} \ar[r]^{f} & Y} consisting of a space X and a continuous map f: A → Y defined on the closed subspace A ⊂ X. Let R be the smallest equivalence relation on X ⨿ Y such that aRf(a) for all a ∈ A; the equivalence classes of R are {a} ⨿ f(a) for a ∈ A, {x} for x ∈ X − A, anf f^{−1}(y) ⨿ {y} for y ∈ Y. The adjunction space is the quotient space

X ∪_{f} Y = X ⨿ Y/R

for the equivalence relation R. Let \overline{f}: X → X ∪_{f} Y be the map X → X ⨿ Y → X ∪_{f} Y and let \overline{i}: Y → X ∪_{f} Y be the map Y → X ⨿ Y → X ∪_{f} Y. These two continuous maps agree on A in the sense that \overline{f} ∘ i = \overline{i} ∘ f and the adjunction space is the universal space with this property. For any other space Z receiving maps X → Z ← Y that agree on A there exists a unique continuous map X ∪_{f} Y such that the diagram

\xymatrix{A \ar@{^{(}->}[d]^{i} \ar[r]^{f} & Y \ar[d]^{\overline{i}} \ar[ddr] \\ X \ar[r]^{\overline{f}} \ar[drr] & X ∪_{f} Y \ar@{.>}[dr]^{∃!} \\  &  & Z}

commutes. The map \overline{i}: Y → X ∪_{f} Y is closed for closed sets B ⊂ Y ⊂ X ⨿ Y have closed saturations f^{−1}(B) ⨿ B. Since \overline{i} is injective it is an embedding; its image is a closed subspace of X ∪_{f} Y homeomorphic to Y. The map \overline{f}|X − A: X − A → X ∪_{f} Y is open for open sets U ⊂ X −A ⊂ X ⨿Y have open saturations U ⨿∅. Since \overline{f}|X −A is injective it is an embedding; its image is an open subspace of X ∪_{f} Y homeomorphic to X −A. The quotient map X ⨿Y → X ∪_{f} Y is closed if the map f is closed for then also closed subsets B ⊂ X ⊂ X ⨿Y have closed saturations B ∪ f^{−1}f(B ∩ A) ⨿ f(A ∩ B). We shall later see [8, Ex 35.8] that X ∪_{f} Y is normal when X and Y are normal.

Only few topological properties are preserved by quotient maps. The reason is that surjective open maps and surjective closed maps are quotient maps so that any property invariant under quotient maps must also be invariant under both open and closed maps.

As we saw in 14.17.(6) the quotient space of a Hausdorff space need not be Hausdorff, not even T_{1}. In general, the quotient space X/R is T_{1} if and only if all equivalence classes are closed sets. (For instance, if X and Y are T_{1} then also the adjunction space X ∪_{f} Y is T_{1}.) The quotient space X/R is Hausdorff if and only if any two distinct equivalence classes are contained in disjoint open saturated sets. We record an easy criterion for Hausdorfness even though you may not yet know the meaning of all the terms.

Proposition 14.19. If X is regular and A ⊂ X is closed then the quotient space X/A is Hausdorff.

The product of two quotient maps need not be a quotient map [2, I.§5.3] in general but here is an important case where it actually is the case.

Theorem 14.20. Let p: A → B and q: C → D be quotient maps. If B and C are locally compact Hausdorff spaces (21.6) then p × q: A × C → B × D is a quotient map.

Proof. Using Lemma 14.21 below we see that the map p × q is the composition

\xymatrix{A × C \ar[r]^{p×1} & B × C \ar[r]^{1×q} & B × D}

of two quotient maps and therefore itself a quotient map. □

Lemma 14.21 (Whitehead Theorem). [5, 3.3.17] Let p: X → Y be a quotient map and Z a locally compact space. Then

p × 1: X × Z → Y × Z

is a quotient map.

Proof. Let A ⊂ X × Z. We must show: (p × 1)^{−1}(A) is open ⟹ A is open. This means that for any point (x, y) ∈ (p × 1)^{−1}(A) we must find a saturated neighborhood U of x and a neighborhood V of y such that U × V ⊂ (p × 1)^{−1}(A).

Since (p × 1)^{−1}(A) is open in the product topology there is a neighborhood U_{1} of x and a neighborhood V of y such that U_{1} × V ⊂ (p × 1)^{−1}(A). Since Y is locally compact Hausdorff we may assume (21.6) that \overline{V} is compact and U_{1} × \overline{V} ⊂ (p × 1)^{−1}(A). Note that also p^{−1}(pU_{1}) × \overline{V} is contained in (p × 1)^{−1}(A). The tube lemma 18.10 says that each point of p^{−1}(pU_{1}) has a neighborhood such that the product of this neighborhood with \overline{V} is contained in the open set (p × 1)^{−1}(A). Let U_{2} be the union of these neighborhoods. Then p^{−1}(pU_{1}) ⊂ U_{2} and U_{2} × \overline{V} ⊂ (p × 1)^{−1}(A). Continuing inductively we find open sets U_{1} ⊂ U_{2} ⊂ ⋯ ⊂ U_{i} ⊂ U_{i+1} ⊂ ⋯ such that p^{−1}(pU_{i}) ⊂ U_{i+1} and U_{i+1} × \overline{V} ⊂ (p × 1)^{−1}(A). The open set U = ⋃ U_{i} is saturated because U ⊂ p^{−1}(pU) = ⋃ p^{−1}(pU_{i}) ⊂ ⋃ U_{i+1} = U. Thus also U ×V is saturated and U ×V ⊂ ⋃ U_{i} ×V ⊂ (p × 1)^{−1}(A). □

For instance, if p: X → Z is a quotient map, then also p × 1: X × [0, 1] → Z × [0, 1] is a quotient map. This fact is important for homotopy theory.

15. Metric topologies

If X is a set with a metric d: X × X → [0, ∞) the collection {B_{d}(x, ε) | x ∈ X, ε > 0} of balls B_{d}(x, ε) = {y ∈ X | d(x, y) < ε} is a basis for the metric topology T_{d} induced by d.

Definition 15.1. A metric space is the topological space associated to a metric set. A topological space is metrizable if the topology is induced by some metric on X.

Hausdorff dimension, fractals, or chaos are examples of metric, rather than topological, concepts.

Theorem 15.2 (Continuity in the metric world). Let f: X → Y be a map between metric spaces with metrics d_{X} and d_{Y}, respectively. The following conditions are equivalent:

(1) f is continuous

(2) ∀x ∈ X∀ε > 0∃δ > 0∀y ∈ X: d_{X}(x, y) < δ ⇒ d_{Y} (f(x), f(y)) < ε

(3) ∀x ∈ X∀ε > 0∃δ > 0: f(B_{X}(x, δ)) ⊂ B_{Y} (f(x), ε)

Proof. Essentially [8, Ex 18.1]. □

Proposition 15.3 (Comparison of metric topologies). Let d and d′ be metrics on X and T_{d}, T_{d′} the associated metric topologies. Then

T_{d} ⊂ T_{d′} ⇔ ∀x ∈ X∀ε > 0∃δ > 0: B_{d′}(x, δ) ⊂ B_{d}(x, ε)

Proof. T_{d} ⊂ T_{d′} if and only if the identity map (X, T_{d′}) → (X, T_{d}) is continuous [8, Ex 18.3] . □

Lemma 15.4 (Standard bounded metric). Let d be a metric on X. Then d′(x, y) = min{d(x, y), 1} is a bounded metric on X (called the standard bounded metric corresponding to d) that induce the same topology on X as d.

Proof. Either use the above proposition or use that for any metric the collection of balls of radius < 1 is a basis for the metric topology. These bases are the same for the two metrics. □

Theorem 15.5 (Hereditary properties of metrizable spaces). Any subspace of a metrizable space is a metrizable. Any countable product of metrizable spaces is metrizable.

Proof. See [8, Ex 21.1] for the first assertion. To prove the second assertion, let X_{n}, n ∈ Z_{+}, be a countable collection of metric spaces. We may assume that each X_{n} has diameter at most 1 (15.4). Put

d((x_{n}), (y_{n})) = sup\{\frac{1}{n}d_{n}(x_{n}, y_{n}) | n ∈ Z_{+}\}

for points (x_{n}) and (y_{n}) of ∏ X_{n} and convince yourself that d is a metric. The idea here is that \frac{1}{n}d_{n}(x_{n}, y_{n}) ≤ \frac{1}{n} becomes small when n becomes large. For any ε > 0

d((x_{n}), (y_{n})) ≤ ε ⟺ ∀n ≤ N: \frac{1}{n}d_{n}(x_{n}, y_{n}) ≤ ε

where N is such that Nε > 1.

The claim is that the metric toplogy coincides with the product topology on ∏ X_{n}. We need to show that the metric topology enjoys the two properties that characterizes the product topology (13.14). First, the projection maps π_{n}: ∏ X_{n} → X_{n} are continuous because d(x, y) < ε ⟹ d_{n}(x, y) < nε (15.2). Second, let f: X → ∏ X_{n} be a map such that X ⟶^{f} ∏ X_{n} ⟶^{π_{n}} X_{n} is continuous for all n. Given x ∈ X and ε > 0, there exist neighborhoods U_{n} of x such that d_{n}(π_{n}f(x), π_{n}f(y)) < nε for all y ∈ U_{n}. Then d(f(x), f(y)) < ε for all y ∈ U_{1} ∩ … ∩ U_{N} where Nε > 1 (remember that all the spaces X_{n} have diameter at most 1). This shows that f: X → ∏ X_{n} is continuous. □

15.6. The first countability axiom. Which topological spaces are metrizable? To address this question we need to build up an arsenal of metrizable and non-metrizable spaces and to identify properties that are common to all metrizable spaces. Here are the first such properties: All metric spaces are Hausdorff and first countable.

Definition 15.7 (Neighborhood basis). A neighborhood basis at x ∈ X is a collection of neighborhoods of x such that any neighborhood of x contains a member of the collection.

Definition 15.8 (First countable spaces). Let X be a space and x a point in X. We say that X has a countable basis at x if there is a countable neighborhood basis at x. X is first countable if all points of X have a countable neighborhood basis.

All metrizable spaces are first countable since {B(x, 1/n) | n ∈ Z_{+}} is a countable neighborhood basis at x.

Proposition 15.9 (Hereditary properties of first countable spaces). [8, Thm 30.2] Any subspace of a first countable space is first countable. Any countable product of first countable spaces is first countable.

Proof. The first assertion is immediate. Let ∏ X_{n} be a countable product of first countable spaces. Let (x_{n}) be a point of ∏ X_{n}. Let B_{n} be countable basis at x_{n} ∈ X_{n}. The collection of all products ∏ U_{n} where U_{n} ∈ B_{n} for finitely many n and U_{n} = X_{n} for all other n is then a countable (5.5) basis at (x_{n}). □

Example 15.10. (1) R is first countable because it is a metric space.

(2) R_{ℓ} is first countable. The collection of half-open intervals [a, b) where b > a is rational is a countable basis of neighborhoods at the point a. Is R_{ℓ} metrizable? (22.3) [8, Ex 30.6]

(3) Is R_{K} first countable? Is it metrizable?

(4) The ordered space \overline{S}_{Ω} = [0, Ω] (6.8) is not first countable at Ω but it is first countable at any other point. Let {U_{n}} be any countable collection of neighborhoods of Ω. Choose a_{n} < Ω such that (a_{n}, Ω] ⊂ U_{n} and chose a such that a_{n} < a < Ω for all n. This is possible because the countable set {a_{n}} ⊂ [0, Ω) of left end-points has an upper bound in [0, Ω) (6.8.(2)). Then (a, Ω] is a neighborhood of Ω that does not contain any of the U_{n} because it does not even contain any of the intervals (a_{n}, Ω]. The section S_{Ω}, however, is first countable: {1} = [1, 2) is open so S_{Ω} has a finite basis at the first element 1. For any other element, α > 1, we can use the countable collection of neighborhoods of the form (β, α] for β < α (12.2.(5)). Is S_{Ω} metrizable? (20.3)[8, Exmp 3 p 181, Ex 30.7].

(5) R^{J} is not first countable at any point when J is uncountable: Let {U_{n}}_{n∈Z_{+}} be any countable collection of neighborhoods of, say, the point (0)_{j∈J} of R^{J}. I claim that there is a neighborhood that does not contain any of the U_{n}. This is because there is an index j_{0} ∈ J such that π_{j_{0}}(U_{n}) = R for all n. Indeed, the set of js for which this is not true

{j ∈ J | ∃n ∈ Z_{+} : π_{j}(U_{n}) ≠ R} = ⋃_{n∈Z_{+}}{j ∈ J | π_{j}(U_{n}) ≠ R}

is a countable union of finite sets, hence countable (5.5.(3)). Then the neighborhood π_{j_{0}}^{−1}(R − {1}) of (0)_{j∈J} does not contain any of the neighborhoods U_{n} in the countable collection. (R^{J} does not even satisfy the sequence lemma (15.12.(1)) [8, Exmp 2 p 133].)

(6) The closed (14.17.(4)) quotient map R → R/Z takes the first countable space R to a space that is not first countable [5, 1.4.17] at the point corresponding to Z. This can be seen by a kind of diagonal argument: Let {U_{n}}_{n∈Z} be any countable collection of open neighborhoods of Z ⊂ R. Let U be the open neighborhood of Z such that U ∩ (n, n + 1), n ∈ Z, equals U_{n} ∩ (n, n + 1) with one point deleted. Then U does not contain any of the U_{n}.

(7) The quotient map p: ∐_{n∈Z_{+}} S^{1} → ⋁_{n∈Z_{+}} S^{1} is closed (14.17.(4)). The domain ∐_{n∈Z_{+}} S^{1} = Z_{+} × S^{1} ⊂ R × R^{2} is first countable (15.9) but the image ⋁_{n∈Z_{+}} S^{1} (14.17.(7)) is not: Let {U_{n}}_{n∈Z_{+}} be any collection of saturated neighborhoods of Z_{+} × {1}. The saturated neighborhood U which at level n equals U_{n} with one point deleted (cf Cantor’s diagonal argument (5.7)) does not contain any of the U_{n}. It follows (15.9) that ⋁_{n∈Z_{+}} S^{1} does not embed in R^{2} nor in any other first countable space. For instance, the universal property of quotient spaces (14.16) gives a factorization

(15.11)  \xymatrix{∐_{n∈Z_{+}} S^{1} \ar[rr]^{f} \ar[dr]_{p} &  & ∏_{n∈Z_{+}} S^{1} \\  & ⋁_{n∈Z_{+}} S^{1} \ar[ur]_{\overline{f}}}

of the continuous map f such that π_{m}(f|{n} × S^{1}) is the identity function when m = n and the constant function when m ≠ n. The induced map \overline{f} is an injective continuous map. It can not be an embedding for the countable product ∏ S^{1} of circles is first countable (15.9). The topology on ⋁ S^{1} is finer than the subspace topology inherited from ∏ S^{1}.

These examples show that the uncountable product of first countable (even metric) spaces and the quotient of a first countable space may fail to be first countable (15.10.(5), 15.10.(7)). Some linearly ordered spaces are first countable, some are not (15.10.(1), 15.10.(4)).

In a first countable (eg metric) space X, the points of the closure of any A ⊂ X can be approached by sequences from A in the sense that they are precisely the limit points of convergent sequences in A. This is not true in general (15.10.(4)).

Lemma 15.12. Let X be a topological space.

(1) (The sequence lemma) Let A ⊂ X be a subspace and x a point of X. Then

x is the limit of a sequence of points from A ⟹ x ∈ \overline{A}

The converse holds if X is first countable.

(2) (Continuous map preserve convergent sequences) Let f: X → Y be a map of X into a space Y. Then

f is continuous ⟹ f(x_{n}) → f(x) whenever x_{n} → x for any sequence x_{n} in X

The converse holds if X is first countable.

Proof. (1) The direction ⟹ is clear. Conversely, suppose that x ∈ \overline{A}. Let U_{n} be a countable basis at x. We may assume that U_{1} ⊃ U_{2} ⊃ ⋯ ⊃ U_{n} ⊃ U_{n+1} ⊃ ⋯ as we may replace U_{n} by U_{1} ∩ … ∩ U_{n}. For each n choose a point x_{n} ∈ A ∩ U_{n}. We claim that the sequence (x_{n}) converges to x. Let U be any neighborhood of x. Then U_{n} ⊂ U for some n so that x_{m} ∈ U_{m} ⊂ U_{n} ⊂ U for m ≥ n.

(2) The direction ⟹ is clear. Conversely, suppose that f(x_{n}) → f(x) whenever x_{n} → x. We want to show that f is continuous, ie (13.3.(6)) that f(\overline{A}) ⊂ \overline{f(A)} for any A ⊂ X. Let x ∈ \overline{A}. Since X is first countable, there is by (1) a sequence of points a_{n} ∈ A converging to x. By hypothesis, the sequence f(a_{n}) ∈ f(A) converges to f(x). Thus f(x) ∈ \overline{f(A)} by (1) again. □

We say that X satisfies the sequence lemma^{1} if for any A ⊂ X and for any x ∈ \overline{A} there is a sequence of points in A converging to x.

To summarize:

X is metrizable ⟹ X is 1st countable ⟹ X satisfies the sequence lemma (is Frechét)

Examples show that neither of these arrows reverse.

The largest element Ω is a limit point of [0, Ω) but it is not the limit of any sequence in [0, Ω) as any such such sequence has an upper bound in [0, Ω) (6.8.(2)). Thus [0, Ω] does not satisfy the sequence lemma. Hnece it is not 1st countable and not metrizable.

15.13. The uniform metric. Let Y be a metric space with a bounded metric d and let J be a set. We shall discuss uniform convergence which is a metric, not a topological, concept.

Definition 15.14. [8, pp 124, 266] The uniform metric on Y^{J} = ∏_{j∈J} Y = map(J, Y) is the metric given by \overline{d}(f, g) = sup\{j ∈ J | d(f(j), g(j))\}.

It is easy to see that this is indeed a metric.

Theorem 15.15. On R^{J} we have: product topology ⊂ uniform topology ⊂ box topology

Proof. Omitted. □

The elements of Y^{J} are functions f: J → Y. Note that a sequence of functions f_{n}: J → Y converges in the uniform metric to the function f: J → Y if and only if

∀ε > 0∃N > 0∀j ∈ J∀n ∈ Z_{+} : n > N ⟹ d(f_{n}(j), f(j)) < ε

We say that the sequence of functions f_{n}: J → Y converges uniformly to the function f: J → Y.

1Aka a Frechét space

Theorem 15.16 (Uniform limit theorem). (Cf [8, Thm 43.6]) Suppose that J is a topological space and that Y is a metric space. The uniform limit of any sequence (f_{n}) of continuous functions f_{n}: J → Y is continuous.

Proof. Well-known. □

16. Connected spaces

Definition 16.1. The topological space X is connected if it is not the union X = X_{0} ∪ X_{1} of two disjoint open non-empty subsets X_{0} and X_{1}.

Two subsets A and B of a space X are separated if A ∩ \overline{B} = ∅ = \overline{A} ∩ B. This means that the two sets are disjoint and neither contains a limit point of the other. Two disjoint open (closed) sets are separated. If C ⊂ A and D ⊂ B and A and B are separated, then C and D are separated. A separation of X consists of two separated non-empty subsets A and B with union X = A ∪ B.

Theorem 16.2. The following are equivalent:

(1) X is connected

(2) The only clopen (closed and open) subsets of X are ∅ and X

(3) X has no separations

(4) Every continuous map X → {0, 1} to the discrete space {0, 1} is constant.

Proof. (1) ⟹ (2): If C is a closed and open subset of X then also the complement X −C is closed and open. Thus X = C ∪ (X − C) is the union of two disjoint open subsets and therefore one of these sets must be empty.

(2) ⟹ (3): Suppose that X = A ∪ B where A and B are separated. Then \overline{A} ⊂ A for \overline{A} does not meet B. Thus A is closed. Similarly, B is closed. Therefore A is both closed and open. By hypothesis, A is either the empty set or all of X.

(3) ⟹ (4): Suppose that there exists a surjective continuous map f: X → {0, 1}. Then X = f^{−1}(0) ∪ f^{−1}(1) is a separation of X.

(4) ⟹ (1): We prove the contrapositive. Suppose that X is non-connected. Then X = U_{1} ∪ U_{2} where U_{1} and U_{1} are disjoint, open, and non-empty. The map f: X → {0, 1} given by f(U_{1}) = 0 and f(U_{1}) = 1 is a non-constant continuous map. □

Theorem 16.3. If X is connected, then f(X) is connected for any continuous map f: X → Y.

Proof. We use the equivalence of (1) and (4) in Theorem 16.2. Let f: X → Y be a continuous map. If f(X) is not connected, there is a non-constant continuous map f(X) → {0, 1} and hence a non-constant continuous map X → {0, 1}. So X is not connected. □

Example 16.4. (1) R − {0} = R_{−} ∪ R_{+} is not connected for it is the union of two disjoint open non-empty subsets.

(2) We shall later prove that R is connected (17.3) and that the connected subsets of R are precisely the intervals, rays, R, and ∅.

(3) R_{ℓ} is not connected for [a, b) is a closed and open subset whenever a < b (12.2.(4)). In fact, any subset Y of R_{ℓ} containing at least two points a < b is disconnected as Y ∩ [a, b) is closed and open but not equal to ∅ or Y. (R_{ℓ} is totally disconnected.)

(4) R_{K} is connected [8, Ex 27.3].

(5) Q is totally disconnected (and not discrete): Let Y be any subspace of Q containing at least two points a < b. Choose an irrational number t between a and b. Then Y ∩(t, ∞) = Y ∩[t, ∞) closed and open but not equal to ∅ or Y.

(6) Particular point topologies (8.2.(3)) are connected.

A subspace of X is said to be connected if it is connected in the subspace topology. A subspace of a connected space need obviously not be connected. So how can we tell if a subspace is connected?

Lemma 16.5. Let Y ⊂ X be a subspace. Then

Y is connected ⟺ Y is not the union of two separated non-empty subsets of X Proof. Suppose that Y = Y_{1} ∪ Y_{2} is the union of two subspaces. Observe that

(16.6)  Y_{1} and Y_{2} are separated in Y ⟺ Y_{1} and Y_{2} are separated in X

because Cl_{Y} (Y_{1}) ∩ Y_{2} =^{12.7.(3)} \overline{Y_{1}} ∩ Y ∩ Y_{2} = \overline{Y_{1}} ∩ Y_{2}. The lemma now follows immediately from 16.2, the equivalence of (1) and (3). □

Corollary 16.7. Suppose that Y ⊂ X is a connected subspace. For every pair A and B of separated subsets of X such that Y ⊂ A ∪ B we have either Y ⊂ A or Y ⊂ B.

Proof. The subsets Y ∩ A and Y ∩ B are separated (since the bigger sets A and B are separated) with union Y. By 16.5, one of them must be empty, Y ∩ A = ∅, say, so that Y ⊂ B. □

Corollary 16.8. The closure of any connected subspace is connected. Indeed, if C ⊂ Y ⊂ \overline{C} and C is connected, then Y is connected.

Proof. Let f: Y → {0, 1} be a continuous map. Then f(C) is a single point since C is connected. Moereover, f(y) = f(C) for all y < inY since otherwise we could separate y and C by disjoint open sets. Thus f is constant. □

Theorem 16.9. Let (Y_{j})_{j∈J} be an indexed family of connected subspaces of the space X. Suppose that there is an index j_{0} ∈ J such that Y_{j} and Y_{j_{0}} are not separated for any j ∈ J. Then the union ⋃_{j∈J} Y_{j} is connected.

Proof. Let f: ⋃_{j∈J} Y_{j} → {0, 1} be a continuous map. The image f(Y_{j}) of Y_{j} is a single point for each j ∈ J since Y_{j} is connected. In fact, f(Y_{j}) = f(Y_{j_{0}}) as otherwise we could separate Y_{j} and Y_{j_{0}} by disjoint open sets. Thus f is constant. □

Corollary 16.10. The union of a collection of connected subspaces with a point in common is connected.

Proof. Apply 16.9 with any of the subspaces as Y_{j_{0}}. □

Corollary 16.11. Suppose that for any two points in X there is a connected subspace containing both of them. Then X is connected.

Proof. Let x_{0} be some fixed point of X. For each point x ∈ X, let C_{x} be a connected subspace containing x_{0} and x. Then X = ⋃ C_{x} is connected as ⋂ C_{x} ≠ ∅ (16.10). □

Theorem 16.12. Products of connected spaces are connected.

Proof. We prove first that the product X × Y of two connected spaces X and Y is connected. In fact, for any two points (x_{1}, y_{1}) and (x_{2}, y_{2}) the subspace X × {y_{1}} ∪ {x_{2}} × Y contains the two points and this subspace is connected since (16.10) it is the union of two connected subspaces with a point in common. Thus X × Y is connected by Corollary 16.11.

Next, induction shows that the product of finitely many connected spaces is connected.

Finally [8, Ex 23.10], consider an arbitrary product ∏ X_{j} of connected spaces X_{j}, j ∈ J. Choose (3.4) a point x_{j} in each of the spaces X_{j} (assuming that all the spaces of the product are nonempty). For every finite subset F of J let C_{F} ⊂ ∏ X_{j} be the product of the subspaces X_{j} if j ∈ F and {x_{j}} if j ∉ F. Since C_{F} is connected for each finite subset F and these subsets have the point (x_{j})_{j∈J} in common, the union ⋃_{F ∈F} C_{F}, where F is the collection of all finite subsets of J, is connected (16.10). This union is not all of ∏ X_{j} but its closure is, so \overline{⋃_{F ∈F} C_{F}} = ∏_{j∈J} X_{j} is connected (16.8). □

17. Connected subsets of linearly ordered spaces

The aim of this section is to determine the connected subsets of R, in fact, more generally, of any linear continuum.

Recall that a subset C of a linearly ordered set X is convex in X if a, b ∈ C ⟹ [a, b] ⊂ C. It is easy to see that connected subspaces are convex. But are all convex subspaces also connected? The convex subset [0, 1] is not connected in Z but it is connected in R.

Lemma 17.2. Suppose that X is a linear continuum (1.15). Then

{connected subsets of X} = {convex subsets of X}

The inclusion ⊂ holds in any linearly ordered space.

Proof. Suppose first that X is any linearly ordered space and let C ⊂ X a subspace that is not convex. Then there exist points a < x < b in X such that a and b are in C and x is outside C. Since C ⊂ (−∞, x) ∪ (x, +∞) is contained in the union of two separated subsets and meet both of them, C is not connected (16.7).

Let now X be a linear continuum and C a convex subset of X. We claim that C is connected. Pick a fixed point a ∈ C and note that C is a union of closed intervals with a as one of their end-points. Therefore it suffices (16.10) to prove the claim in case C = [a, b] is a closed interval. Suppose that [a, b] = A ∪ B is the union of two disjoint relative open sets A and B where b ∈ B. We must show that A = ∅ (16.5). The sets A and B are closed and open in [a, b]. As the closed interval [a, b] is closed in X (9.1), A and B are also closed in X (12.7). If we assume, in order to arrive at a contradiction, that A is nonempty, then the nonempty bounded set A has a least upper bound, c = sup A. Then c ≤ b since b is an upper bound and c ∈ A since A is closed in X. (The least upper bound of any bounded set always belongs to the closure of the set since otherwise it wouldn’t be the least upper bound.) So c < b (for b ∉ A) and A ⊂ [a, b). Now, A is open in [a, b] and in the subspace [a, b) (which has the subspace topology which is the order topology (11.6)). But this contradicts c ∈ A for no element of an open subset A of [a, b) can be an upper bound for A: For any point d ∈ A there exists an open interval (x, y) around d such that [a, b) ∩ (x, y) ⊂ A and since [d, y) ≠ ∅, d is not an upper bound for A. (In short, c ∈ A since A is closed in X and c ∉ A since A is open in [a, b).) This is a contradiction and therefore A must be empty. □

The inclusion ⊃ does not hold for all linearly ordered spaces as for instance Z_{+} is convex but not connected (there are gaps).

In particular, the connected subsets of R are precisely the convex subsets which are ∅, R, and all intervals (bounded or unbounded, closed, open or halfopen).

We shall now identify the linearly ordered spaces that are connected.

Theorem 17.3. [2, Ex 7 p 382, Prop 1 p 336] Let X be a linearly ordered space. Then

X is connected in the order topology ⟺ X is a linear continuum

The connected subsets of a linear continuum X are: X, ∅, intervals and rays.

Proof. ⟹: [8, Ex 24.4] [5, Problem 6.3.2] [2, Ex 7 p 382]. Suppose that X is a linearly ordered set that is not a linear continuum. Then there are nonempty, proper, clopen subsets of X:

• If (x, y) = ∅ for some points x < y then (−∞, x] = (∞, y) is clopen and ≠ ∅, X.

• If A ⊂ X is a nonempty subset bounded from above which has no least upper bound then

the set of upper bounds B = ⋂_{a∈A}[a, ∞) = ⋃_{b∈B}(b, ∞) is clopen and ≠ ∅, X. Therefore X is not connected (16.2). ⟸: Assume X is a linear continuum. From 17.2 we know that the the connected and the convex subsets of X are the same. In particular, the linear continuum X, certainly convex in itself, is connected in the order topology. Let C be a nonempty convex subset of X. We look at two cases:

• C is neither bounded from above nor below. Let x be any point of X. Since x is neither a lower nor an upper bound for C there exist a, b ∈ C so that a < x < b. Then x ∈ C by convexity. Thus C = X.

• C is bounded from above but not from below. Let c = sup C be its least upper bound. Then C ⊂ (−∞, c]. Let x < c be any point. Since x is neither a lower nor an upper bound for C there exist a, b ∈ C so that a < x < b. Then x ∈ C by convexity. Thus (∞, c) ⊂ C ⊂ (−∞, c] and C is either (−∞, c) or (−∞, c].

The arguments are similar for the other cases. Recall that X also has the greatest lower bound property [8, Ex 3.13]. □

The real line R, the ordered square I_{o}^{2} (9.4.(7)), the (ordinary) (half) line [1, ω) = Z_{+} × [0, 1) (6.5.(6)), and the long (half) line [0, Ω) × [0, 1) (6.8) [8, Ex 24.6, 24.12] are examples of linear continua.

Theorem 17.4. [Intermediate Value Theorem] Let f: X → Y be a continuous map of a connected topological space X to a linearly ordered space Y. Then f(X) is convex. If Y is a linear continuum, f(X) is an interval (bounded or unbounded, closed, half-open, or open)

Proof. X connected ⟹^{(16.3)} f(X) connected ⟹^{(17.3)} f(X) convex. For subsets of a linear continuum we know (17.3) that connected = convex = interval. □

Any linearly ordered space containing two consecutive points, two points a and b with (a, b) = ∅ is not connected as X = (−∞, b) ∪ (a, +∞) is the union of two disjoint open sets.

Any well-ordered set X containing at least two points is totally disconnected in the order topology. For if C ⊂ X contains a < b then a ∉ C ∩ (a, b] ∋ b is closed and open in C since (a, b] is closed and open in X.

17.5. Path connected spaces. Path connectedness is a stronger property than connectedness.

Definition 17.6. The topological space X is path connected if for any two points x_{0} and x_{1} in X there is a continuous map (a path) f: [0, 1] → X with f(0) = x_{0} and f(1) = x_{1}.

The image under a continuous map of a path connected space is path connected, cf 16.3. Any product of path connected spaces is path connected [8, Ex 24.8], cf 16.9.

Example 17.7. The punctured euclidian plane R^{n} − {0} is path connected when n ≥ 2 since any two points can be joined by a path of broken lines. Thus also the (n − 1)-sphere S^{n−1}, which is the image of R^{n} − {0} under the continuous map x ↦ x/|x|, is path connected for (n − 1) ≥ 1.

Since the unit interval I is connected (17.3), all paths f(I) are also connected (16.3) so all path connected spaces are, as unions of paths emanating from one fixed point, connected (16.11). The converse is not true, not all connected spaces are path connected.

Example 17.8. (1) (Topologist’s sine curve) Let S be the graph of the function sin(1/x), 0 < x ≤ 1, considered as a subspace of the plane R^{2}. The (closed) topologist’s sine curve is the subspace \overline{S} = S ∪(\{0\}×[−1, 1]) of the plane. It follows from 16.3 and 16.8 that \overline{S} is connected. We shall soon see that \overline{S} is not path connected (17.17.(2)).

(2) The ordered square I_{o}^{2} (9.4.(7)) is connected since it is a linear continuum [8, Ex 3.15] but it is not path connected. For suppose that f: [0, 1] → I_{o}^{2} is a path from the smallest element (the lower left corner) f(0) = 0 × 0 to the largest element (the upper right corner) f(1) = 1 × 1. Then f is surjective for the image contains (17.4) the interval [0 × 0, 1 × 1] = I_{o}^{2}. But this is impossible since the ordered square I_{o}^{2} contains uncountably many open disjoint subsets (eg x×(0, 1) = (x×0, x×1), 0 ≤ x ≤ 1) but [0, 1] does not contain uncountably many open disjoint subsets (choose a rational number in each of them).

(3) R_{K} (8.10) is connected but not path-connected space [8, Ex 27.3].

(4) The Stone–Čech compactification βR of R is connected but not path-connected [8, Ex 38.9].

17.9. Components and path components. The relations on X defined by

x ∼ y ⇔ There is a connected subset C ⊂ X such that x ∈ C and y ∈ C
x ∼^{p} y ⇔ There is a path in X between x and y

are equivalence relations. Check transitivity! (You may need the Glueing lemma (13.5.(6)).)

Definition 17.10. The components of X are the equivalence classes of the first equivalence relation. The path components of X are the equivalence classes of the second equivalence relation

The component containing the point x ∈ X is, by definition, the union of all connected subspaces containing x. The path component of the point x ∈ X consists, by definition, of all points with a path to x.

Theorem 17.11. (1) The components are connected closed disjoint subsets with union X. Any connected subset of X is contained in precisely one of the components.

(2) If X = ⋃ C_{α}, where the C_{α} are connected subspaces such that for all α ≠ β there is a separation A ∪ B of X with C_{α} ⊂ A and C_{β} ⊂ B, then the C_{α} are the components of X.

(3) The path components of X are path connected disjoint subsets with union X. Any path connected subset is contained in precisely one of the path components.

(4) The path components are connected. The components are unions of path components.

Proof. (1) The components form a partition of X since they are equivalence classes (1.11). The components are connected by the definition of the relation ∼ and 16.11. All the points of a connected subset are equivalent so they are contained in the same equivalence class. In particular (16.11), \overline{C} ⊂ C for any component C so C is closed.

(2) Since each C_{α} is connected, it is contained in precisely one component (17.11.(1)). Could there be two of the C_{α}s, say C_{α} and C_{β}, inside one component, say C? No: Choose a separation X = A ∪ B such that C_{α} ⊂ A and C_{β} ⊂ B. Then C ⊂ X = A ∪ B and C meets both A and B which is impossible (16.7).

(3) The path components are path-connected by their very definition. All path-connected spaces are connected, so each path-component is contained in precisely one component. Thus the components are unions of path-components. □

If X has only finitely many components the components are closed and open. If X = C_{1}∪⋯∪C_{n} where the finitely many subspaces C_{i} are connected and separated then the C_{i} are the components of X.

The path components can be closed, open, closed and open, or neither closed nor open. However, for locally connected spaces (17.13) components and path components coincide (17.15).

17.12. Locally connected and locally path connected spaces.

Definition 17.13. The space X is locally (path) connected at the point x if every neighborhood of x contains a (path) connected neighborhood of x. The space X is locally (path) connected if it is locally (path) connected at each of its points.

Thus a space is locally (path) connected iff it has a basis of (path) connected subsets.

Consider this table

R R − \{0\} \overline{S}  Q
Connected YES NO YES NO
Locally connected YES YES  NO NO

before you draw any conclusions about the relationship between connected and locally connected spaces. Note in particular that a space can be connected (even path connected (17.17)) and not locally connected.

On the positive side, note that open subsets of locally (path) connected spaces are locally (path) connected and that quotient spaces of locally connected spaces are locally connected (17.16).

Proposition 17.14. The space X is locally (path) connected if and only if open subsets have open (path) components (open in X or, what is the same (11.3), open in the open set). In a locally (path-) connected space the (path-) components are clopen.

Proof. Assume that X is locally connected and let U be an open subset. Consider a component, C, of U. The claim is that C is open. Let x be a point in C. Choose a connected neighborhood V of x such that V ⊂ U. Since V is connected and intersects C, V ⊂ C (17.11). This shows that C is open. In particular, the components (of the open set X) are open and, since they form a partition, also closed.

Conversely, assume that open subsets have open components. Let x be a point of X and U a neighborhood of x. Let C be the component of U containing x. Then C is an open connected neighborhood of x contained in U. □

It follows for instance that the open subsets of R are unions of at most countably many open intervals. For any open set is the union of its components which are open, since R is locally connected, and connected, so they are open intervals. Since each of the open intervals contains a rational number, there are at most countably many components. Any closed subset of R is the complement to a union of at most countably many open intervals.

Theorem 17.15. In a locally path-connected space the path-components and the components are the same.

Proof. Suppose that X is locally path-connected. Each path-component P is contained (17.11) in a unique component, C. Since C is connected and P is clopen (17.14), P = C (16.2). □

For instance, the components and the path components of a locally Euclidean space, such as a manifold, are the same. A locally path-connected space is path-connected if and only if it is connected.

Proposition 17.16. [8, Ex 25.8] [2, I.§11.6] Locally (path) connected spaces have locally (path) connected quotient spaces.

Proof. Let X be a locally connected space and p: X → Y a quotient map. Let V be a subset of Y and C a component of V. Then p^{−1}(C) is a union of components of p^{−1}(V) because continuous maps preserve connectedness (16.3). If V is open in Y, p^{−1}(V) is open in X, and since X is locally connected it follows (17.14) that p^{−1}(C) is open in X. This means (14.6) that C is open in Y. Since thus open subsets of Y have open components, Y is locally connected (17.14). (The same proof applies with ‘connected’ replaced by ‘path connected’.) □

Example 17.17. (1) The comb space X = ([0, 1]×{0})∪(({0}∪{1/n | n ∈ Z_{+}})×[0, 1]) ⊂ R^{2} is clearly path connected but it is not locally connected for (17.15) the open set U = X−[0, 1]×{0} has a component (use (17.11.(2)) to identify the components) that is not open. (The same is true for any (small) neighborhood of (0, 1/2).) See [8, Ex 25.5] for a similar example.

(2) The closed topologist’s sine curve \overline{S} is not locally connected: Let U be a small neighborhood (well, not too big) around (0, 1/2). Then U has a component that is not open. For instance, U = \overline{S} −([0, 1]×\{0\} has a component (17.11.(2)), \{0\}×(0, 1] that is not open (no neighborhood of (0, ±\frac{1}{2}) is disjoint from S). It follows (17.16, 18.8.(2)) that \overline{S} is not path-connected for it can not be a quotient space of the locally connected compact space [0, 1]. In fact, \overline{S} has two pathcomponents: S = \overline{S}∩(R_{+}×R), which is open and not closed for its closure is \overline{S}, and \{0\}×[0, 1], which is closed and not open for S is not closed. Since there are two path components and one component, \overline{S} is not locally path connected (17.15). (The Warsaw circle, the union of \overline{S} with an arc from one end of S to [0, 1]) is an example of a path connected not locally connected space, so is the next example.)

(3) The space X = \overline{⋃ C_{n}} (12.13.(2)) is even path connected but still not locally connected for the component R_{+} × {0} of the open set X − {(0, 0)} (use (17.11.(2)) to identify the components) is not open (any neighborhood of (1, 0) contains points from the circles). Alternatively, X is not locally connected since any small neighborhood of (1, 0) has a separation.

(4) Any linear continuum is locally connected since the basis for the topology consists of connected sets (17.3). It need not be locally path connected [8, Ex 25.3].

(5) The (generalized) Jordan curve theorem says that R^{n} − h(S^{n−1}) has two path-components for any embedding h: S^{n−1} → R^{n} of an (n − 1)-sphere into R^{n}, n ≥ 1. This is usually proved using algebraic topology.

englishGENERAL TOPOLOGY 47

Figure 3. The Warsaw circle

18. Compact spaces

Definition 18.2. A topological space X is compact if it satisfies one of the following two equivalent conditions:

(1) Every collection of open subsets whose union is X contains a finite subcollection whose union is X.

(2) Every collection of closed subsets whose intersection is ∅ contains a finite subcollection whose intersection is ∅.

A collection of open subsets whose union is the whole space is called an open covering. Thus a space is compact if any open covering contains a finite open covering.

Any finite space is compact as there are only finitely many open subsets. Any compact discrete space is finite. The real line R is not compact as the open covering {(−n, n) | n ∈ Z_{+}} has no finite subcovering.

How can we recognize the compact subspaces of a given topological space?

Proposition 18.3. Let Y be a subspace of X. The following are equivalent:

(1) Y is compact

(2) Every collection of open subsets of X whose union contains Y contains a finite subcollection whose union contains Y

(3) Every collection of closed subsets of X whose intersection is disjoint from Y contains a finite subcollection whose intersection is disjoint from Y

Proof. (1) ⟹ (2): Let {U_{j} | j ∈ J} be a collection of open subsets of X such that Y ⊂ ⋃ U_{j}. Then {Y ∩ U_{j} | j ∈ J} is an open covering of Y. Since Y is compact, Y = ⋃_{j∈J′} Y ∩ U_{j} for some finite index set J′ ⊂ J. This means that Y ⊂ ⋃_{j∈J′} U_{j}.

(2) ⟸ (1): Let {V_{j} | j ∈ J} be an open covering of Y. Then V_{j} = Y ∩ U_{j} for some open set U_{j} ⊂ X and Y ⊂ ⋂_{j∈J} U_{j}. By assumption, Y ⊂ ⋃_{j∈J′} U_{j} for some finite index set J′ ⊂ J. Therefore Y = Y ∩ ⋃_{j∈J′} U_{j} = ⋃_{j∈J′} Y ∩ U_{j} = ⋃_{j∈J′} V_{j}. This shows that Y is compact.

(2) ⟺ (3): Clear from DeMorgan’s laws. □

Theorem 18.4. Closed subspaces of compact spaces are compact.

Proof. Suppose that X is a compact space and Y ⊂ X a closed subset. Let F_{j}, j ∈ J, be a collection of closed subsets of X such that Y ∩ ⋂_{j∈J} F_{j} = ⋂_{j∈J} Y ∩ F_{j} = ∅. By compactness of X, ∅ = ⋂_{j∈J′} Y ∩ F_{j} = Y ∩ ⋂_{j∈J′} F_{j} for some finite index set J′ ⊂ J. Thus Y is compact by 18.3. □

Theorem 18.5. If X is compact, then f(X) is compact for any continuous map f: X → Y.

Proof. Let U_{j}, j ∈ j, be a collection of open subsets of Y that cover the image f(X). Then X = ⋃_{j∈J} f^{−1}(U_{j}) is an open covering of X. Since X is compact, X = ⋃_{j∈J′} f^{−1}(U_{j}) for some finite index set J′ ⊂ J. Then f(X) = f( ⋃_{j∈J′} f^{−1}(U_{j})) = ⋃_{j∈J′} ff^{−1}(U_{j}) ⊂ ⋃_{j∈J′} U_{j}. This shows that f(X) is compact (18.3). □

The compact subspaces of a Hausdorff space have some of the same properties as points.

Theorem 18.6. (1) Compact subspaces of Hausdorff spaces are closed.

(2) Any two disjoint compact subspaces of a Hausdorff space can be separated by disjoint open sets.

Proof. Let L and K be two disjoint compact subspaces of X. Consider first the special case where L = {x_{0}} is a point. For each point x ∈ K, the Hausdorff property implies that there are disjoint open sets U_{x}, V_{x} such that and x ∈ U_{x} and x_{0} ∈ V_{x}. Since K is compact (18.3), K ⊂ U_{x_{1}} ∪…∪U_{x_{t}} for finitely many points x_{1}, …, x_{t} ∈ K. Set U = U_{x_{1}} ∪ … ∪ U_{x_{t}} and V = V_{x_{1}} ∩ … ∩ V_{x_{t}}. The existence of V alone says that K is closed.

Assume next that L is any compact subspace of X. We have just seen that for each point y ∈ L there are disjoint open sets U_{y} ⊃ K and V_{y} ∋ y. By compactness, L is covered by finitely many of the V_{y}. Then K is contained in the intersection of the corresponding finitely many U_{y}. □

Corollary 18.7. Let X be a compact Hausdorff space.

(1) Let C ⊂ X a subset. Then C is compact ⟺ C is closed.

(2) If A and B are disjoint closed sets in X then there exist disjoint open sets U, V such that A ⊂ U and B ⊂ V.

Proof. (1) ⟹ is 18.6 and ⟸ is 18.4.

(2) Let A and B be disjoint closed subsets of X. Then A and B are compact as just shown. Now apply 18.6. □

Lemma 18.8 (Closed Map Lemma). Suppose that X is a compact space, Y is a Hausdorff space, and f: X → Y is a continuous map.

(1) f is closed (14.1).

(2) If f is surjective, it is a closed quotient map (14.8).

(3) If f is injective, it is an embedding (13.10).

(4) If f is bijective, it is a homeomorphism (13.7).

Proof. Let f: X → Y be a continuous map of a compact space X into a Hausdorff space Y.

(1) We have

C is closed in X ⟹^{18.4} C is compact ⟹^{18.5} f(C) is compact ⟹^{18.6} f(C) is closed in Y

which shows that f is a closed map.

(2) Every closed continuous surjective map is a closed quotient map (14.12).

(3) Every closed continuous injective map is an embedding (13.3).

(4) Every closed bijective continuous map is a homeomorphism (14.10).

□

Corollary 18.9. If f: X → Y is continuous and X is compact and locally connected and Y is Hausdorff, then f(X) is compact and locally connected.

Proof. We know (18.8) that f: X → f(X) is a (closed) quotient map. The quotient image of a locally connected space is locally connected (17.16). □

Lemma 18.10 (The Tube Lemma). Let X and Y be topological spaces where Y is compact. Let x_{0} be a point of X. For any neighborhood N ⊂ X × Y of the slice {x_{0}} × Y there is a neighborhood U of x_{0} such that {x_{0}} × Y ⊂ U × Y ⊂ N.

Proof. For each point y ∈ Y there is a product neighborhood such that x_{0} × y ⊂ U_{y} × V_{y} ⊂ N. Since Y is compact, there are finitely many points y_{1}, …, y_{k} ∈ Y such that Y = V_{1} ∪…∪V_{k} where V_{i} = V_{y_{i}}. Put U = U_{1} ∩ … U_{k}. □

Theorem 18.11. (Cf 18.16) The product of finitely many compact spaces is compact.

Proof. We show that the product X ×Y of two compact spaces X and Y is compact. The theorem in its general form follows by induction. Suppose that A_{j}, j ∈ J, is an open covering of X ×Y. We show that there is a finite subcovering. For each point x ∈ X, the slice {x} × Y, homeomorphic to Y, is compact and therefore {x} × Y ⊂ ⋃_{j∈J_{x}} A_{j} for some finite index set J_{x} ⊂ J (18.3). Thanks to the tube lemma (18.10) we know that the open set ⋃_{j∈J_{x}} A_{j} actually contains a whole tube U_{x} × Y for some neighborhood U_{x} of x. By compactness, X = U_{x_{1}} ∪ … ∪ U_{x_{k}} can be covered by finitely many of the neighborhoods U_{x}. Now

X × Y = ( ⋃_{1≤i≤k} U_{x_{i}}) × Y = ⋃_{1≤i≤k}(U_{x_{i}} × Y) = ⋃_{1≤i≤k} ⋃_{j∈J_{x_{i}}} A_{j} = ⋃ ; j∈⋃_{1≤i≤k} J_{x_{i}} A_{j}

where ⋃_{1≤i≤k} J_{x_{i}} ⊂ J is finite (4.6). □

Here are two small lemmas that are used quite often.

Lemma 18.12 (Criterion for noncompactness). If X contains an infinite closed discrete subspace, then X is not compact.

Proof. Any closed discrete subspace of a compact space is compact (18.4) and discrete, hence finite. □

Lemma 18.13 (Intersection of a nested sequence of compact sets). [8, Ex 28.5] [5, 3.10.2] Let C_{1} ⊃ C_{2} ⊃ ⋯ ⊃ C_{n} ⊃ ⋯ be a descending sequence of closed nonempty subsets of a compact space. Then ⋂ C_{n} ≠ ∅.

Proof. If ⋂ C_{n} = ∅ then C_{n} = ∅ for some n ∈ Z_{+} by 18.2.(2). □

Theorem 18.14. [Cf [8, Ex 27.5]] A nonempty compact Hausdorff space without isolated points (12.11) is uncountable.

Proof. Let x_{n} be a sequence of points in X. It suffices (5.3) to show that {x_{n} | n ∈ Z_{+}} ≠ X. Since X is Hausdorff and has no isolated points there is a descending sequence

V_{1} ⊃ V_{2} ⊃ ⋯ ⊃ V_{n−1} ⊃ V_{n} ⊃ ⋯

of nonempty open sets such that x_{n} ∉ \overline{V}_{n} for all n. These are constructed recursively. Put V_{0} = X. Suppose that V_{n−1} has been constructed for some n ∈ Z_{+}. Since x_{n} is not isolated, {x_{n}} ≠ V_{n−1}. Choose a point y_{n} ∈ V_{n−1} distinct from x_{n} and choose disjoint (separated) open sets U_{n} ⊂ X, V_{n} ⊂ V_{n−1} such that x_{n} ∈ U_{n} and y_{n} ∈ V_{n}. Then U_{n} ∩ \overline{V}_{n} = ∅ so x_{n} ∉ \overline{V}_{n}.

By construction, the intersection ⋂ \overline{V}_{n} contains none of the points \{x_{n}\} of the sequence and by compactness (18.13), ⋂ \overline{V}_{n} ≠ ∅. Thus X contains a point that is not in the sequence. □

The Alexandroff compactification (21.3) ωZ_{+} = \overline{K} = K ∪ \{0\}, where K is as in 8.10, is a countable compact Hausdorff space with isolated points.

Is it true that a connected Hausdorff space is uncountable?

Example 18.15. (1) The (unreduced) suspension of the space X is the quotient space

SX = [0, 1] × X/({0} × X, {1} × X)

What is the suspension of the n-sphere S^{n}? Define f: [0, 1] × S^{n} → S^{n+1} to be the (continuous) map that takes (t, x), t ∈ [0, 1], to the geodesic path from the north pole (0, 1) ∈ S^{n+1}, through the equatorial point (x, 0) ∈ S^{n} ⊂ S^{n+1}, to the south pole (0, −1) ∈ S^{n+1}. (Coordinates in R^{n+2} = R^{n+1} × R.) By the universal property of quotient spaces there is an induced continuous and bijective map \overline{f}: SS^{n} → S^{n+1}. Since SS^{n} is compact (as a quotient of a product of two compact spaces) and S^{n+1} is Hausdorff (as a subspace of a Hausdorff space), \overline{f} is a homeomorphism (18.8). We often write SS^{n} = S^{n+1}, n ≥ 0, where the equality sign stands for ‘is homeomorphic to’.

(2) Every injective continuous map S^{1} → R^{2} is an embedding. Can you find an injective continuous map R^{1} → R^{2} that is not an embedding?

(3) Alexander’s horned sphere [7, Example 2B.2] is a wild embedding of the 2-sphere S^{2} in R^{3} such that the unbounded component of the complement R^{3} − S^{2} (17.17.(5)) contains noncontractible loops. It may be easier instead to consider Alexander’s horned disc [3, p 232]. (The horned sphere is obtained by cutting out a disc of the standard sphere and replacing it a horned disc.)

(4) Let X be a Hausdorff space and

X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n−1} ⊂ X_{n} ⊂ ⋯ ⊂ X

an ascending sequence of closed subspaces. Assume that the topology on X is coherent with this filtration in the sense that

A is closed ⟺ A ∩ X_{n} is closed for all n

holds for all subsets A of X. Then any compact subspace C of X is contained in a finite stage of the filtration. To see this, choose a point t_{n} ∈ C ∩ (X_{n} − X_{n−1}) for all n ∈ Z_{+} for which this intersection is nonempty. Let T = {t_{n}} be the set of these points. We want to prove that T is finite. Certainly, T ∩ X_{n} is finite, hence closed in the Hausdorff space X_{n} for all n ≥ 0. Therefore T is closed. In fact, any subset of T is closed by the same argument and thus T is discrete. But a closed discrete subspace of a compact space is finite (18.12).

Theorem 18.16 (Tychonoff theorem). The product ∏_{j∈J} X_{j} of any collection (X_{j})_{j∈J} of compact spaces is compact.

Proof. Put X = ∏_{j∈J} X_{j}. Let us say that a collection of subsets of X is an FIP-collection (finite intersection property) if any finite subcollection has nonempty intersection. It is enough (18.2.(2)) to show that

A is FIP ⟹ ⋂_{A∈A} \overline{A} ≠ ∅

holds for any collection A of subsets of X.

So let A be a FIP-collection iof subsets of X. Somehow we must find a point x in ⋂_{A∈A} \overline{A}. Step 1. The FIP-collection A is contained in a maximal FIP-collection. The set (supercollection?) 𝔸 of FIP-collections containing A is a strictly partially ordered by strict inclusion. Any linearly ordered subset (subsupercollection?) 𝔹 ⊂ 𝔸 has an upper bound, namely the FIP-collection ⋃ 𝔹 = ⋃_{B∈𝔹} B containing A. Now Zorn’s lemma (7.4) says that 𝔸 has maximal elements.

We can therefore assume that A is a maximal FIP-collection. Step 2. Maximality of A implies

(18.17)  A_{1}, …, A_{k} ∈ A ⟹ A_{1} ∩ ⋯ ∩ A_{k} ∈ A

and for any subset A_{0} ⊂ X we have

(18.18)  ∀A ∈ A: A_{0} ∩ A ≠ ∅ ⟹ A_{0} ∈ A

To prove (18.17), note that the collection A∪{A_{1} ∩⋯∩A_{k}} is FIP, hence equals A by maximality. To prove (18.18), note that the collection A ∪ {A_{0}} is FIP: Let A_{1}, …, A_{k} ∈ A. By (18.17), A_{1} ∩ ⋯ ∩ A_{k} ∈ A so A_{1} ∩ ⋯ ∩ A_{k} ∩ A ≠ ∅ by assumption. Step 3. {π_{j}(A) | A ∈ A} is FIP for all j ∈ J. π_{j}(A_{1}) ∩ ⋯ ∩ π_{j}(A_{k}) ⊃ π_{j}(A_{1} ∩ ⋯ ∩ A_{j}) ≠ ∅ for any finite collection of sets A_{1}, …, A_{k} ∈ A.

Since X_{j} is compact and \{\overline{π_{j}(A)} | A ∈ A\} is an FIP collection of closed subsets of X_{j}, the intersection ⋂_{A∈A} \overline{π_{j}(A)} ⊂ X_{j} is nonempty (18.2.(2)). Choose a point x_{j} ∈ ⋂ \overline{π_{j}(A)} for each j ∈ J and put x = (x_{j}) ∈ X. Step 3. x ∈ ⋂_{A∈A} \overline{A}. The point is that π_{j}^{−1}(U_{j}) ∈ A for any neighborhood U_{j} of x_{j}. This follows from (18.18) since π_{j}^{−1}(U_{j}) ∩ A ≠ ∅, or equivalently, U_{j} ∩ π_{j}(A) ≠ ∅, for any A ∈ A as x_{j} ∈ \overline{π_{j}(A)}. Since A has the FIP it follows that π_{j_{1}}^{−1}(U_{j_{1}}) ∩ ⋯ ∩ π_{j_{k}}^{−1}(U_{j_{k}}) ∩ A ≠ ∅ whenever U_{j_{i}} are neighborhoods of the finitely many points x_{j_{i}} ∈ X_{j_{i}} and A ∈ A. Thus x ∈ \overline{A} for all A in the collection A; in other words x ∈ ⋂_{A∈A} \overline{A}. □

19. Compact subspaces of linearly ordered spaces

We show first that any compact subspace of a linearly ordered space is contained in a closed interval.

Lemma 19.1. Let X be a linearly ordered space and ∅ ≠ C ⊂ X a nonempty compact subspace. Then C ⊂ [m, M] for some elements m, M ∈ C.

Proof. The claim is that C has a smallest and a largest element. The proof is by contradiction. Assume that C has no largest element. Then C ⊂ ⋃_{c∈C}(−∞, c) as there for any a ∈ C is a c ∈ C such that a < c. By compactness

C ⊂ (−∞, c_{1}) ∪ ⋯ ∪ (−∞, c_{k}) ⊂ (−∞, c)

where c = max{c_{1}, …, c_{k}} is the largest (6.6) of the finitely many elements c_{1}, …, c_{k} ∈ C. But this is a contradiction as c ∈ C and c ∉ (∞, c). □

We now show that the eg the unit interval [0, 1] is compact in R. Note that the unit interval [0, 1] is not compact in R−\{\frac{1}{2}\}. (The intersection all the closed subspaces [−\frac{1}{n} + \frac{1}{2}, \frac{1}{n} + \frac{1}{2}] ⊂ [0, 1], n ≥ 2, is empty but the intersection of finitely many of them is not empty.) The reason for this difference is that R, but not R − \{\frac{1}{2}\} (1.16.(6)), has the least upper bound property (1.15).

Theorem 19.2. Let X be a linearly ordered space with the least upper bound property. Then every closed interval [a, b] in X is compact.

Proof. Let [a, b] ⊂ X be a closed interval and A and open covering of [a, b] (with the subspace topology which is the order topology (11.6)). We must show that [a, b] is covered by finitely many of the sets from the collection A. The set

C = {x ∈ [a, b] | [a, x] can be covered by finitely many members of A}

is nonempty (a ∈ C) and bounded from above (by b). Let c = sup C be the least upper bound of C. Then a ≤ c ≤ b. We would like to show that c = b. \underline{Step 1} If x ∈ C and x < b then C ∩ (x, b] ≠ ∅. Proof of Step 1. Suppose first that x has an immediate successor y > x. We can not have x < b < y for then y would not be an immediate successor. So x < y ≤ b. Clearly [a, y] = [a, x] ∪ {y} can be covered by finitely many members of A. Suppose next that x has no immediate successor. Choose an open set A ∈ A containing x. Since A is open in [a, b] and contains x < b, A contains an interval of the form [x, d) for some d ≤ b. Since d is not an immediate successor of x there is a point y ∈ (x, d). Now [a, y] ⊂ [a, x] ∪ [x, d) ⊂ [a, x] ∪ A can be covered by finitely sets from A. \underline{Step 2} c ∈ C. Proof of Step 2. The claim is that [a, c] can be covered by finitely many members of A. From Step 1 we have that C contains elements > a so the upper bound c is also > a. Choose A ∈ A such that c ∈ A. Since A is open in [a, b] and a < c, A contains an interval of the form (d, c] for some d where a ≤ d. Since d is not an uppe bound for C, there are points from C in (d, c]. Let y be such a point. Now [a, c] = [a, y] ∪ (d, c] ⊂ [a, y] ∪ A can be covered by finitely many sets from A. \underline{Step 3} c = b. Proof of Step 3. By Step 2, c ∈ C. But then c = b for Step 1 says that if c < b, then c can not be an upper bound. □

For instance, the ordered square I_{o}^{2} = [0 × 0, 1 × 1], \overline{Z_{+}} = [1, ∞], and \overline{S}_{Ω} = [1, Ω] are compact linearly ordered spaces [8, Ex 10.1]. More importantly, since the linearly ordered space R has the least upper bound property, we get that all closed intervals in R are compact.

See [8, Ex 27.1] for the converse.

Corollary 19.3. Let X be a linearly ordered space and ∅ ≠ C ⊂ X a nonempty subspace. Then

C is compact and connected ⟹ C is a closed interval

The converse holds when X is a linear continuum.

Proof. Since C is compact, C ⊂ [m, M] for some m, M ∈ C (19.1) and [m, M] ⊂ C because C is connected, hence convex (17.2). In a a linear continuum every closed interval is compact (19.2) and connected (17.2). □

In particular, the compact and connected subsets of the linear continuum R are precisely the closed intervals.

Theorem 19.4 (Heine–Borel). A subset C ⊂ R^{n} is compact if and only if it is closed and bounded in the standard metric.

Proof. Let C ⊂ R^{n} be compact. Since C is compact and {(−R, R)^{n} | R > 0} an open covering of R^{n}, there is an R > 0 such that C ⊂ [−R, R]^{n}. Thus C is bounded. Compact subspaces Hausdorff spaces are closed (18.6), so C is closed.

Conversely, if C is closed and bounded then C is a closed subset of [−R, R]^{n} for some R > 0. But [−R, R]^{n} is compact (19.3, 18.11) and closed subsets of compact spaces are compact (18.4); in particular, C is compact. □

Can you now answer question 13.11.(10)?

Theorem 19.5 (Extreme Value theorem). Let f: X → Y be a continuous map of a nonempty topological space X into a linearly ordered space Y.

(1) If X is compact then there exist m, M ∈ X such that f(m) ≤ f(x) ≤ f(M) for all x ∈ X.

(2) If X is compact and connected then there exist m, M ∈ X such that f(X) = [f(m), f(M)].

Proof. (1) Since the image f(X) is compact (18.5) we can apply (19.1).

(2) Since the image f(X) is compact (18.5) and connected (16.3) we can apply (19.3). □

19.6. Compactness in metric spaces. Since any open covering of compact space contains a finite subcovering we see that any open covering of a compact metric space contains a subcovering of open sets of size bounded from below. This observation is expressed in the Lebesgue lemma.

Lemma 19.7 (Lebesgue lemma). Let X be a compact metric space. For any open covering of X there exists a δ > 0 (the Lebesgue number of the open covering) such that any subset of X with diameter < δ is contained in one of the open sets in the covering.

Proof. Cover X by finitely many open balls, X = B(x_{1}, δ_{1}) ∪ B(x_{2}, δ_{2}) ∪ ⋯ ∪ B(x_{k}, δ_{k}), such that each double radius ball B(x_{i}, 2δ_{i}) is contained in one of the open sets of the covering. Let δ be any positive number smaller than all the δ_{i}. Then any subset of diameter < δ is contained in one of the balls B(x_{i}, 2δ_{i}) and therefore in one of the open sets from the covering. (Let A ⊂ X be a subset with diam(A) < δ. Choose any point a in A. Then a is in one of the open balls B(x_{i}, δ_{i}). For any point b ∈ A, the distance d(b, x_{i}) ≤ d(b, a) + d(a, x_{i}) < δ + δ_{i} < 2δ_{i} which means that the point b is contained in B(x_{i}, 2δ_{i}).) □

The open covering of R consisting of the open intervals (−1, 1) and (x − 1/|x|, x + 1/|x|) for all x ∈ R with |x| > 1 does not have a Lebesgue number.

Definition 19.8. A map f: X → Y between metric spaces is uniformly continuous if for all ε > 0 there is a δ > 0 such that d_{Y} (f(x_{1}), f(x_{2})) < ε whenever d_{X}(x_{1}, x_{2}) < δ:

∀ε > 0∃δ > 0∀x_{1}, x_{2} ∈ X: d_{X}(x_{1}, x_{2}) < δ ⇒ d_{Y} (f(x_{1}), f(x_{2})) < ε

Theorem 19.9 (Uniform continuity theorem). Let f: X → Y be a continuous map between metric spaces. If X is compact, then f is uniformly continuous.

Proof. Given ε > 0. Let δ be the Lebesgue number of the open covering \{f^{−1}B(y, \frac{1}{2}ε) | y ∈ Y \}. Then we have for all x_{1}, x_{2} ∈ X

d_{X}(x_{1}, x_{2}) < δ ⟹ diam{x_{1}, x_{2}} < δ

⟹ ∃y ∈ Y : \{x_{1}, x_{2}\} ⊂ f^{−1}B(y, \frac{1}{2}ε)

⟹ ∃y ∈ Y : \{f(x_{1}), f(x_{2})\} ⊂ B(y, \frac{1}{2}ε)

⟹ d_{Y} (f(x_{1}), f(x_{2})) < ε

□

20. Limit point compactness and sequential compactness

Definition 20.1. A space X is (1) limit point compact if any infinite subset of X has a limit point

(2) sequentially compact if any sequence in X has a convergent subsequence.

In other words, a space is limit point compact if it contains no infinite closed discrete subspaces (12.12).

We shall not say much about these other forms of compactness (see [5, §3.10] for a more thorough discussion).

Theorem 20.2. (Cf [8, Ex 28.4, Ex 35.3]) For any topological space X we have

X is compact ⟹ X is limit point compact ⟹^{X 1st countable + T_{1}} X is sequentially compact

All three forms of compactness are equivalent for a metrizable space.

Proof. X is compact ⟹ X is limit point compact: A subset with no limit points is closed and discrete (12.12), hence finite (18.12). X is limit point compact, 1st countable and T_{1} ⟹ X is sequentially compact: Let (x_{n}) be a sequence in X. Consider the set A = {x_{n} | n ∈ Z_{+}} of points in the sequenec. If A is finite there is a constant subsequence (4.6.(3)). If A is infinite, A has a limit point x by hypothesis. If X is 1st countable, there is a countable nested countable basis, U_{1} ⊃ U_{2} ⊃ ⋯, at x just as in the proof of (15.12). Since x is a limit point, there is a sequence element x_{n_{1}} ∈ U_{1}. Suppose that we have found n_{1} < ⋯ < n_{k} such that x_{n_{i}} ∈ U_{i}. Since x is a limit point and X is T_{1}, there are infinitely many points from A in U_{k+1} (12.20). In particular, there is n_{k+1} < n_{k} such that x_{n_{k+1}} is in U_{k+1}. The subsequence x_{n_{k}} converges to x. X is sequentially compact and metrizable ⟹ X compact: This is more complicated (but should also be well-known from your experience with metric spaces) so we skip the proof. □

Example 20.3. (1) The well-ordered space S_{Ω} of all countable ordinals is limit point compact and sequentially compact but it is not compact. S_{Ω} is not compact (19.1) for it is a linearly ordered space with no greatest element [8, Ex 10.6]. On the other hand, any countably infinite subset of S_{Ω} is contained in a compact subset (6.8.(2), 19.2). Therefore (20.2) any countably infinite subset, indeed any infinite subset (12.12), has a limit point and any sequence has a convergent subsequence. (Alternatively, use that S_{Ω} is first countable (15.10.(4)).) It follows (20.2) that S_{Ω} is not metrizable.

(2) The Stone–Čech compactification (26.10) βZ_{+} of the positive integers is compact but it is not limit point compact as the sequence (n)_{n∈Z_{+}} has no convergent subsequence. Indeed, no point of the remainder βZ_{+} − Z_{+} is the limit of a sequence in Z_{+} [8, Ex 38.9]. (It follows that βZ_{+} is not first countable and not metrizable).

21. Locally compact spaces and the Alexandroff compactification

Definition 21.1. A space X is locally compact at the point x ∈ X if x lies in the interior of some compact subset of X. A space is locally compact if it is locally compact at each of its points.

This means that X is locally compact at the point x if x ∈ U ⊂ C where U is an open and C a compact subset of X.

Compact spaces are locally compact (U = X = C). The real line R (more generally, R^{n}) is locally compact but not compact. All linearly ordered spaces with the least upper bound property are locally compact. The space Q ⊂ R of rational numbers is not locally compact [8, Ex 29.1].

Definition 21.2. A compactification of the space X is an embedding c: X → cX of X into a compact Hausdorff space cX such that c(X) is dense in cX.

If X itself is compact then X = cX as X is closed and dense in cX. So compactifications are only interesting for noncompact spaces.

Let X be a locally compact Hausdorff space. Let ωX = X ∪ {ω} denote the union of X with a set consisting of a single point ω. The collection

T = {U | U ⊂ X open} ∪ {ωX − C | C ⊂ X compact}

is a topology on ωX: It is easy to see that finite intersections and arbitrary unions of open sets of the first (second) kind are again open of the first (second) kind. It follows that T is closed under arbitrary unions. It is also closed under finite intersections since U ∩ (ωX − C) = U − C is open in X (18.6). And ωX = ωX − ∅ and ∅ are open.

If X is compact, then ωX is X with an added isolated point. The next theorem describes ωX in the more interesting case when X is not compact.

Theorem 21.3 (Alexandroff). Let X be a locally compact but not compact Hausdorff space. (1) The map X → ωX is a compactification missing one point.

(2) If c: X → cX is another compactification of X then there is a unique closed quotient map \underline{c}: cX → ωX such that the diagram

\xymatrix{ & X \ar[dl]^{c} \ar[dr] \\ cX \ar[rr]_{\underline{c}} &  & ωX}

commutes.

Proof. We must verify the following points: The subspace topology on X is the topology on X: The subspace topology X ∩ T is clearly the original topology on X. X is dense in ωX: The intersection of X and some neighborhood of ω has the form X − C where C is compact. Since X is assumed non-compact, X − C is not empty. ωX is Hausdorff: Let x_{1} and x_{2} be two distinct points in ωX. If both points are in X then there are disjoint open sets U_{1}, U_{2} ⊂ X ⊂ ωX containing x_{1} and x_{2}, respectively. If x_{1} ∈ X and x_{2} = ω, choose an open set U and a compact set C in X such that x ∈ U ⊂ C. Then U ∋ x and ωX −C ∋ ω are disjoint open sets in ωX. ωX is compact: Let {U_{j}}_{j∈J} be any open covering of ωX. At least one of these open sets contains ω. If ω ∈ U_{k}, k ∈ J, then U_{k} = (X − C) ∪ {ω} for some compact set C ⊂ X. There is a finite set K ⊂ J such that {U_{j}}_{j∈K} covers C. Then {U_{j}}_{j∈K∪{k}} is a finite open covering of ωX = X ∪{ω}.

Let now c: X → cX be another compactification X. Define \underline{c}: cX → ωX by \underline{c}(x) = x for all x ∈ X and \underline{c}(cX − X) = ω. We check that \underline{c} is continuous. For any open set U ⊂ X ⊂ ωX, \underline{c}^{−1}(U) is open in X and hence (11.3.(2)) in cX since X is open in cX (21.4). For any compact set C ⊂ X, \underline{c}^{−1}(ωX − C) = cX − C is open in cX since the compact set C is closed in the Hausdorff space cX (18.7.(1)). This shows that \underline{c} is continuous. By construction, \underline{c} is surjective and hence (18.8) a closed quotient map by the Closed Map Lemma (18.8). □

The theorem says that ωX = cX/(cX − X) where cX is any compactification of X. In particular, if cX consists of X and one extra point then the map \underline{c} is a bijective quotient map, ie a homeomorphism (14.10). The space ωX is called the Alexandroff compactification or the one-point compactification of X.

Lemma 21.4. Any locally compact and dense subspace of a Hausdorff space is open.

Proof. Suppose that Y is Hausdorff and that the subspace X ⊂ Y is locally compact and dense. Let x ∈ X. Since X is locally compact Hausdorff the point x has a neighborhood X ∩ U, where U is open in Y, such that its relative closure

Cl_{X}(X ∩ U) {12.7 \atop =} X ∩ \overline{X ∩ U} {12.9 \atop =} X ∩ \overline{U}

is compact and hence closed in the Hausdorff space Y (18.6). But no part of U can stick outside X ∩ \overline{U} for since U − (X ∩ \overline{U}) is open

U − (X ∩ \overline{U}) ≠ ∅ ⟹^{X dense} (X ∩ U) − (X ∩ \overline{U}) ≠ ∅ ⚡

which is absurd. Thus we must have U ⊂ X ∩ \overline{U}, in particular, U ⊂ X. This shows that X is open. □

Example 21.5. (1) If X is compact Hausdorff then ωX is X together with an isolated point ω. ({ω} = ωX − X is a neighborhood of ω.)

(2) The n-sphere S^{n} = ωR^{n} is the Alexandroff compactification of the locally compact Hausdorff space R^{n} for there is (13.11.(8)) a homeomorphism of R^{n} onto the complement of a point in S^{n}.

(3) Let X be a linearly ordered space with the least upper bound property and let [a, b) be a half-open interval in the locally compact space X. Then ω[a, b) = [a, b]. For instance the Alexandroff compactifications of the half-open intervals [0, 1) ⊂ R, Z_{+} = [1, ω) ⊂ Z_{+} × Z_{+}, and S_{Ω} = [1, Ω) ⊂ \overline{S}_{Ω} are [0, 1], [1, ω], [1, Ω] = \overline{S}_{Ω}, respectively.

(4) ωZ_{+} ∧ S^{1} = ω(Z_{+} × R) = ⋃_{n∈Z_{+}} C_{1/n}, the Hawaiian Earring (12.13.(2)) (which is compact by the Heine-Borel theorem (19.4)).

(5) The Warsaw circle W (which compact by the Heine-Borel theorem (19.4)) (17.17.(2)) is a compactification of R and W/(W − R) = ωR = S^{1}.

Corollary 21.6 (Characterization of locally compact Hausdorff spaces). In the category of Hausdorff spaces, the following conditions are equivalent:

(1) X is locally compact

(2) X is homeomorphic to an open subset of a compact space

(3) For any point x ∈ X and any neighborhood U of x there is an open set V such that x ∈ V ⊂ \overline{V} ⊂ U and \overline{V} is compact.

Proof. (1) ⟹ (2): The locally compact Hausdorff space X is homeomorphic to the open subspace ωX − {ω} of the compact Hausdorff space ωX (21.3).

(2) ⟹ (3): Suppose that Y is a compact Hausdorff space and that X ⊂ Y is an open subset. X is Hausdorff since subspaces of Hausdorff spaces are Hausdorff (12.19). Let x be a point of X and U ⊂ X a neighborhood of x in X. Then U is open also in Y (11.3) so that the complement Y − U is closed and compact (18.4) in the compact space Y. In the Hausdorff space Y we can separate (18.7.(2)) the disjoint closed subspaces {x} and Y − U by disjoint open sets, V ∋ x and W ⊃ Y − U or Y − W ⊂ U. As V is disjoint from W it is contained in the closed subset Y − W and then also \overline{V} ⊂ Y − W ⊂ U. Here, \overline{V} is compact as a closed subspace of the compact space Y (18.4). Note (12.7) that the closure of V in X equals the closure of V in Y because \overline{V}, the closure of V in Y, is contained in U and X.

(3) ⟹ (1): Let x be a point of X. By property (3) with U = X there is an open set V such that x ∈ V ⊂ \overline{V} ⊂ X and \overline{V} is compact. Thus X is locally compact at x. □

Corollary 21.7. (1) Closed subsets of locally compact spaces are locally compact. (2) Open or closed subsets of locally compact Hausdorff spaces are locally compact Hausdorff.

Proof. (1) Let X be a locally compact space and A ⊂ X a closed subspace. We use the definition (21.1) directly to show that A is locally compact. Let a ∈ A. There are subsets a ∈ U ⊂ C ⊂ X such that U is open and C is compact. Then A ∩ U ⊂ A ∩ C where A ∩ U is a neighborhood in A and C ∩ A is compact as a closed subset of the compact set C (18.4).

(2) If X is locally compact Hausdorff and A ⊂ X open, it is immediate from 21.6.(2) that A is locally compact Hausdorff. □

An arbitrary subspace of a locally compact Hausdorff space need not be locally compact (21.8.(1)). The product of finitely many locally compact spaces is locally compact but an arbitrary product of locally compact spaces need not be locally compact [8, Ex 29.2]; for instance Z_{+}^{ω} is not locally compact. The image of a locally compact space under an open continuous [8, 29.3] or a perfect map [8, Ex 31.7] [5, 3.7.21] is locally compact but the image under a general continuous map of a locally compact space need not be locally compact; indeed, the quotient of a locally compact space need not be locally compact (21.8.(2)) [5, 3.3.16].

Example 21.8. (1) ⋃_{n∈Z_{+}} C_{n} ⊂ R^{2} (12.13.(2)) is not locally compact at the origin: Any neighborhood of 0 × 0 contains a countably infinite closed discrete subspace so it can not be contained in any compact subspace (18.12).

(2) The quotient space R/Z_{+} (15.10.(6)) is not locally compact at the point corresponding to Z_{+}: Any neighborhood of this point contains an infinite closed discrete subspace so it can not be contained in any compact subspace (18.12).

(3) In diagram (15.11), the space ∐ S^{1} is locally compact, ∏ S^{1} is compact, and ⋁ S^{1} is not locally compact (at the one point common to all the circles).

(4) (Wedge sums and smash products) A pointed space is a topological space together with one of its points, called the base point. The wedge sum of two pointed disjoint spaces (X, x_{0}) and (Y, y_{0}) is the quotient space

X ∨ Y = (X ⨿ Y)/({x_{0}} ∪ {y_{0}})

obtained from the disjoint union of X and Y (10.6) by identifying the two base points. Let f: X ∪ Y → X × Y be the continuous map given by f(x) = (x, y_{0}), x ∈ X, and f(y) = (x_{0}, y), y ∈ Y. The image f(X ∪ Y) = (X × {y_{0}}) ∪ ({x_{0}} × Y) = π_{2}^{−1}({y_{0}}) ∪ π_{1}^{−1}({x_{0}}) is closed when X and Y are T_{1}–spaces. In fact, f is a closed map for if C ⊂ X is closed then f(C) = {(x, y_{0}) | x ∈ C} = π_{1}^{−1}(C) ∩ π_{2}^{−1}({y_{0}}) is also closed. Thus f is a quotient map onto its image and its factorization \overline{f} as in the commutative diagram (14.16.(2))

\xymatrix{X ∪ Y \ar[dr] \ar[rr]^{f} &  & X × Y \\  & X ∨ Y \ar[ur]_{\overline{f}}}

is an embedding. We can therefore identify X ∨Y and X ×{y_{0}}∪{x_{0}}×Y. The smash product is defined to be

X ∧ Y = (X × Y)/(X ∨ Y)

the quotient of the product space X × Y by the closed subspace X ∨ Y (× − ∨ = ∧).

If A ⊂ X and B ⊂ Y are closed subspaces, the universal property of quotient maps (14.16.(2)) produces a continuous bijection \overline{g} such that the diagram

\xymatrix{X × Y \ar[dr] \ar[r]^{g_{X}×g_{Y}} & X/A × Y/B \ar[r]^{g_{X/A×Y /B}} & X/A ∧ Y/B \\  & (X × Y)/(X × B ∪ A × Y) \ar[ur]_{\overline{g}}}

commutes. However, \overline{g} may not be a homeomorphism since the product g_{X} × g_{Y} of the two closed quotient maps g_{X}: X → X/A and g_{Y} : Y → Y/B may not be a quotient map. If X/A and Y are locally compact Hausdorff spaces ([Ex 31.7] may be useful here) then g_{X} × g_{Y} is quotient [Ex 29.11] so that \overline{g} is a homeomorphism (14.11, 14.16.(3), 14.10) and

(21.9)  (X × Y)/(X × B ∪ A × Y) = X/A ∧ Y/B

in this case.

(5) The torus S^{1} × S^{1} is a compactification of R × R with remainder S^{1} ∨ S^{1}.

(6) (The Alexandroff compactification of a product space) If X and Y are locally compact Hausdorff spaces the map X × Y → ωX × ωY → ωX ∧ ωY is an embedding. This follows from (14.14, 14.10) when we note that the first map embeds X × Y into an open saturated subset of ωX × ωY (13.12). The universal property of the Alexandroff compactification (21.3) implies that

(21.10)  ω(X × Y) = ωX ∧ ωY

for any two locally compact Hausdorff spaces X and Y. In particular,

S^{m} ∧ S^{n} = ωR^{m} ∧ ωR^{n} =^{(21.10)} ω(R^{m} × R^{n}) = ωR^{m+n} = S^{m+n}

when we view the spheres as Alexandroff compactifications of euclidian spaces.

(7) The 1–sphere S^{n} is (homeomorphic to) the quotient space I/∂I. Hence

S^{n} = S^{1} ∧ … ∧ S^{1} = I/∂I ∧ … ∧ I∂I = I^{n}/∂I^{n}

using the formula [Exam June 2003] for the boundary of the product of two sets.

Some mathematicians prefer to include Hausdorffness in the definition of (local) compactness. What we call a (locally) compact Hausdorff space they simply call a (locally) compact space; what we call a (locally) compact space they call a (locally) quasicompact space.

22. Countability Axioms

Let us recall the basic question: Which topological spaces are metrizable? In order to further analyze this question we shall look at a few more properties of topological spaces.

We have already encountered the first countability axiom. There are three more axioms using the term ’countable’. Here are the four countability axioms.

Definition 22.1. A topological space • that has a countable neighborhood basis at each of its points is called a first countable space • that has a countable basis is called a second countable space

• that contains a countable dense subset is said to contain a countable dense subset^{4}

• in which every open covering has a countable subcovering is called a Lindelöf space

A subset A ⊂ X is dense if \overline{A} = X, that is if every nonempty open subset contains a point from A.

Any second countable space is first countable. Any subspace of a (first) second countable space is (first) second countable. A countable product of (first) second countable spaces is (first) second countable (15.9). Quotient spaces of first coountable spaces need not be first countable (15.10.(7)).

The real line R is second countable for the open intervals (a, b) with rational end-points form a countable basis for the topology. R_{ℓ} is first but not second countable (23.5).

Metric spaces are first countable but not necessarily second countable [8, Example 2 p 190]. Any compact space is Lindelöf. Any closed subset of a Lindelöf space is Lindelöf [8, Ex 30.9]

(with a proof that is similar to that of 18.4). A product of two Lindelöf spaces need not be Lindelöf (23.6).

The next result implies that R, in fact any subset of the 2nd countable space R, is Lindelöf.

Theorem 22.2. Let X be a topological space. Then

\xymatrix{X has a countable dense subset & X is 2nd countable \ar@{=>}[l] \ar@{=>}[r] \ar@{=>}[d] & X is Lindelöf \\  & X is 1st countable}

If X is metrizable, the three conditions of the top line equivalent.

Proof. Suppose first that X is second countable and let B be a countable basis for the topology. \underline{X has a countable dense subset}: Pick a point b_{B} ∈ B in each basis set. Then \{b_{B} | B ∈ B\} is countable (5.5.(2)) and dense. \underline{X is Lindelöf}: Let U be an open covering of X. For each basis set B ∈ B for which it is possible pick a member U_{B} of the open covering U such that B ⊂ U_{B}. Then the at most countable collection {U_{B}} of these open sets is an open covering: Let x be any point in X. Since x is contained in a member of U and every open set is a union of basis sets we have x ∈ B ⊂ U for some basis set B and some U ∈ U. But then also x ∈ B ⊂ U_{B}. \underline{Any metric space with a countable dense subset is 2nd countable}: Let X be a metric space with a countable dense subset A ⊂ X. Then the collection {B(a, r) | a ∈ A, r ∈ Q_{+}} of balls centered at points in A and with a rational radius is a countable (5.5.(4)) basis for the topology: It suffices to show that for any open ball B(x, ε) in X and any y ∈ B(x, ε) there are a ∈ A and r ∈ Q_{+} such that y ∈ B(a, r) ⊂ B(x, ε). Let r be a positive rational number such that 2r + d(x, y) < ε and let a ∈ A ∩ B(y, r). Then y ∈ B(a, r), of course, and B(a, r) ⊂ B(x, ε) for if d(a, z) < r then d(x, z) ≤ d(x, y) + d(y, z) ≤ d(x, y) + d(y, a) + d(a, z) < d(x, y) + 2r < ε. \underline{Any metric Lindelöf space is 2nd countable}: Let X be a metric Lindelöf space. For each positive rational number r, let A_{r} be a countable subset of X such that X = ⋃_{a∈A_{r}} B(a, r). Then A = ⋃_{r∈Q_{+}} A_{r} is a dense countable (5.5.(3)) subset: For any open ball B(x, ε) and any positive rational r < ε there is an a ∈ A_{r} such that x ∈ B(a, r). Then a ∈ B(x, r) ⊂ B(x, ε). □

Example 22.3. The ordered square I_{o}^{2} is compact (19.2) and therefore Lindelöf but it is not second countable since it contains uncountably many disjoint open sets (x × 0, x × 1), x ∈ I. Thus I_{o}^{2} is not metrizable [8, Ex 30.6].

4Or to be separable

23. Separation Axioms

Definition 23.1. A space X is called a • T_{1}-space if points {x} are closed in X

• T_{2}-space or a Hausdorff space if for any pair of distinct points x, y ∈ X there exist disjoint open sets U, V ⊂ X such that x ∈ U and y ∈ V

• T_{3}-space or a regular space if points are closed and for any point x ∈ X and any closed set B ⊂ X not containing x there exist disjoint open sets U, V ⊂ X such that x ∈ U and B ⊂ V

• T_{4}-space or a normal space if points are closed and for every par of disjoint closed sets A, B ⊂ X there exist disjoint open sets U, V ⊂ X such that A ⊂ U and B ⊂ V.

We have the following sequence of implications

X is normal ⟹ X is regular ⟹ X is Hausdorff ⟹ X is T_{1}

where none of the arrows reverse (23.6, 23.7, [8, Ex 22.6]).

Lemma 23.2. Let X be a T_{1}-space. Then

(1) X is regular ⟺ For every point x ∈ X and every neighborhood U of x there exists an open set V such that x ∈ V ⊂ \overline{V} ⊂ U.

(2) X is normal ⟺ For every closed set A and every neighborhood U of A there is an open set V such that A ⊂ V ⊂ \overline{V} ⊂ U.

Proof. Let B = X − U. □

Theorem 23.3. (Cf 12.19) Any subspace of a regular space is regular. Any product of regular spaces is regular.

Proof. Let X be a regular space and Y ⊂ X a subset. Then Y is Hausdorff (12.19)). Consider a point y ∈ Y and a and a closed set B ⊂ X such that y ∉ Y ∩ B. Then y ∉ B and since Y is regular there exist disjoint open sets U and V such that y ∈ U and \overline{B} ⊂ V. The relatively open sets U ∩ Y and V ∩ Y are disjoint and they contain y and B ∩ Y, respectively.

Let X = ∏ X_{j} be the Cartesian product of regular space X_{j}, j ∈ J. Then X is Hausdorff (12.19)). We use 23.2.(1) to show that X is regular. Let x = (x_{j}) be a point in X and U = ∏ U_{j} a basis neighborhood of x. Put V_{j} = X_{j} whenever U_{j} = X_{j}. Otherwise, choose V_{j} such that x_{j} ∈ V_{j} ⊂ \overline{V}_{j} ⊂ U_{j}. Then V = ∏ V_{j} is a neighborhood of x in the product topology and (13.16) \overline{V} = ∏ \overline{V}_{j} ⊂ ∏ U_{j} = U. Thus X is regular. □

Theorem 23.4. [8, Ex 32.1] A closed subspace of a normal space is normal.

Proof. Quite similar to the proof (23.3) that a subspace of a regular space is regular. □

An arbitraty subspace of a normal space need not be normal (26.5) and the product of two normal spaces need not be normal ((23.6),[8, Example 2 p 203], [5, 2.3.36]).

Example 23.5 (Sorgenfrey’s half-open interval topology). The half-open intervals [a, b) form a basis for the space R_{ℓ} (8.10, 12.2.(4), 15.10.(2), 16.4.(3)).

R_{ℓ} is 1st countable: At the point x the collection of open sets of the form [x, b) where b > x

is rational, is a countable local basis at x. R_{ℓ} is not 2nd countable: Let B be any basis for the topology. For each point x choose a member B_{x} of B such that x ∈ B_{x} ⊂ [x, x + 1). The map R → B: x ↦ B_{x} is injective for if B_{x} = B_{y} then x = inf B_{x} = inf B_{y} = y.

R_{ℓ} has a countable dense subset: Q is dense since any (basis) open set in R_{ℓ} contains

rational points. R_{ℓ} is Lindelöf: It suffices (!) to show that any open covering by basis open sets contains a countable subcovering, ie that if R = ⋃_{j∈J}[a_{j}, b_{j}) is covered by a collection of right half-open intervals [a_{j}, b_{j}) then R is actually already covered by countably many of these intervals. (Note that this is true had the intervals been open as R is Lindelöf.) Write

R = ⋃_{j∈J}(a_{j}, b_{j}) ∪ (R − ⋃_{j∈J}(a_{j}, b_{j}))

as the (disjoint) union of the corresponding open intervals and the complement of this union. The first set can be covered by countably many of the intervals (a_{j}, b_{j}) for any subset of R (with the standard topology) is 2nd countable and hence Lindelöf (22.2). Also the second set can be covered by countably many of the intervals (a_{j}, b_{j}) simply because it is countable: The second set

R − ⋃_{j∈J}(a_{j}, b_{j}) = ⋃_{j∈J}[a_{j}, b_{j}) − ⋃_{j∈J}(a_{j}, b_{j}) = {a_{k} | ∀j ∈ J: a_{k} ∉ (a_{j}, b_{j})}

consists of some of the left end-points of the intervals. The open intervals (a_{k}, b_{k}) are disjoint for a_{k} in this set. But there is only room for countably many open disjoint intervals in R (choose a rational point in each of them) so there are at most countaly many left end-dpoints a_{k} in the second set.

R_{ℓ} is normal: Let A, B ⊂ R_{ℓ} be disjoint closed sets. For each point a ∈ A ⊂ R − B there is an x_{a} ∈ R such that [a, x_{a}) ⊂ R − B. Let U_{A} = ⋃_{a∈A}[a, x_{a}). Define U_{B} = ⋃_{b∈B}[b, x_{b}) similarly. Then U_{A} and U_{B} are open sets containing A and B, respectively. If U_{A} ∩U_{B} ≠ ∅ then [a, x_{a}) ∩ [b, x_{b}) ≠ ∅ for some a ∈ A, b ∈ B. Say a < b; then b ∈ [a, x_{a}) ⊂ R − B which is a contradiction ⚡ So U_{A} and U_{B} are disjoint. (R_{ℓ} is even completely normal [8, Ex 32.6] by this argument.)

Since R_{ℓ} has a dense countable subset and is not second countable (23.5) it is not metrizable (22.2).

Example 23.6 (Sorgenfrey’s half-open square topology). The half-open rectangles [a, b) × [c, d) form a basis (10.3) for the product topology R_{ℓ} × R_{ℓ}. The anti-diagonal L = {(x, −x) | x ∈ R} is a closed (clear!) discrete (L ∩ [x, ∞) × [−x, ∞) = {(x, −x)}) subspace of the same cardinality as R. Q × Q is a countable dense subspace.

R_{ℓ} × R_{ℓ} is not Lindelöf: A Lindelöf space can not contain an uncountable closed discrete

subspace (cf 18.12, [8, Ex 30.9]). R_{ℓ} × R_{ℓ} is not normal: A normal space with a countable dense subset can not contain

a closed discrete subspace of the same cardinality as R [5, 2.1.10]. (Let X be a space with a countable dense subset. Since any continuous map of X into a Hausdorff space is determined by its values on a dense subspace [8, Ex 18.13], the set of continuous maps X → R has at most the cardinality of R^{Z} = ∏_{Z} R. Let X be any normal space and L a closed discrete subspace. The Tietze extension theorem (24.5) says that any map L → R extends to continuous map X → R. Thus the set of continuous maps X → R has at least the cardinality of R^{R} = ∏_{R} R which is greater (5.6) than the cardinality of ∏_{Z} R.) See [8, Ex 31.9] for a concrete example of two disjoint closed subspaces that can not be separated by open sets.

R_{ℓ} × R_{ℓ} is (completely) regular: since it is the product (26.3) of two (compeletely) regular (26.1) (even normal (23.5)) spaces.

Example 23.5 shows that the arrows of Theorem 22.2 do not reverse. Example 23.6 shows that the product of two normal spaces need not be normal, that the product of two Lindelöf spaces need not be Lindelöf, and provides an example of a (completely) regular space that is not normal.

Example 23.7 (A Hausdorff space that is not regular). The open intervals plus the sets (a, b)−K where K = \{\frac{1}{n} | n ∈ Z_{+}\} form a basis for the topology R_{K} (8.10, 17.8.(3)). This is a Hausdorff topology on R since it is finer than the standard topology. But R_{K} is not regular. The set K is closed and 0 ∉ K. Suppose that U ∋ 0 and V ⊃ K are disjoint open sets. We may choose U to be a basis open set. Then U must be of the form U = (a, b) − K for the other basis sets containing 0 intersect K. Let k be a point in (a, b) ∩ K (which is nonempty). Since k is in the open set V, there is a basis open set containing k and contained in V; it must be of the form (c, d). But U ∩ V ⊃ ((a, b) − K) ∩ (c, d) and ((a, b) − K) ∩ (c, d) ≠ ∅ for cardinality reasons. This is a contradiction ⚡

24. Normal spaces

Normal spaces can be characterized in two ways.

Corollary 24.1. Let X be a T_{1}-space. Then the following conditions are equivalent:

(1) X is normal: For any pair of disjoint closed sets A, B ⊂ X there exist disjoint open sets U, V ⊂ X such that A ⊂ U and B ⊂ V.

(2) (Urysohn’s characterization of normality) For any pair of disjoint closed sets A, B ⊂ X there exists a continuous function f: X → [0, 1] such that f(A) = 0 and f(B) = 1.

(3) (Tietze’s characterization of normality) For any closed subset A of X and any continuous function f: A → [0, 1] there exists a continuous function F: X → [0, 1] such that F |A = f.

The proof relies on Urysohn’s lemma (24.2) and Tietze’s extension theorem (24.5).

Theorem 24.2 (Urysohn lemma). Let X be a normal space and let A and B be disjoint closed subsets of X. Then there exists a continuous function (a Urysohn function) f: X → [0, 1] such that f(a) = 0 for a ∈ A and f(b) = 1 for b ∈ B.

Proof. We shall recursively define open sets U_{r} ⊂ X for all r ∈ Q ∩ [0, 1] such that (make a drawing!)

(24.3)  r < s ⟹ A ⊂ U_{0} ⊂ U_{r} ⊂ \overline{U}_{r} ⊂ U_{s} ⊂ \overline{U}_{s} ⊂ U_{1} ⊂ X − B

To begin, let U_{1} be any open set such that A ⊂ U_{1} ⊂ X − B, for instance U_{1} = X − B. By normality (23.2.(2)) there is an open set U_{0} such that A ⊂ U_{0} ⊂ \overline{U}_{0} ⊂ U_{1}. To proceed, arrange the elements of Q ∩ [0, 1] into a sequence

Q ∩ [0, 1] = {r_{0} = 0, r_{1} = 1, r_{2}, r_{3}, r_{4}, …}

such that the first two elements are r_{0} = 0 and r_{1} = 1. Assume that the open sets U_{r_{i}} satisfying condition (24.3) have been defined for i ≤ n, where n ≥ 1. We shall now define U_{r_{n+1}}. Suppose that if we put the numbers r_{0}, r_{1}, …, r_{n}, r_{n+1} in order

0 = r_{0} < ⋯ < r_{ℓ} < r_{n+1} < r_{m} ⋯ < r_{1} = 1

the immediate predecessor of r_{n+1} is r_{ℓ} and r_{m} is the immediate successor. Then \overline{U}_{r_{ℓ}} ⊂ U_{r_{m}} by (24.3). By normality (23.2.(2)) there is an open set U_{r_{n+1}} such that \overline{U}_{r_{ℓ}} ⊂ U_{r_{n+1}} ⊂ \overline{U}_{r_{n+1}} ⊂ U_{r_{m}}. The sets U_{r_{i}}, i ≤ n + 1, still satisfy (24.3).

We are now ready to define the function. Consider the function f: X → [0, 1] given by

f(x) = \begin{cases}inf\{r ∈ Q ∩ [0, 1] | x ∈ U_{r}\} & x ∈ U_{1} \\ 1 & x ∈ X − U_{1} \end{cases}

Then f(B) = 1 by definition and f(A) = 0 since A ⊂ U_{0}. But why is f continuous? It suffices (13.2) to show that the subbasis intervals (9.1) [0, a), a > 0, and (b, 1], b < 1, have open preimages. Since

f(x) < a ⟺ ∃r < a: x ∈ U_{r} ⟺ x ∈ ⋃_{r<a} U_{r}

f(x) > b ⟺ ∃r′ > b: x ∉ U_{r′} ⟺ ∃r > b: x ∉ \overline{U}_{r} ⟺ x ∈ ⋃_{r>b}(X − \overline{U}_{r})

the sets

f^{−1}([0, a)) = ⋃_{r<a} U_{r} and f^{−1}((b, 1]) = ⋃_{r>b}(X − \overline{U}_{r})

are open. □

Example 24.4. Let X = R and let A = (−∞, −1] and B = [2, ∞). If we let U_{r} = (−∞, r) for r ∈ Q ∩ [0, 1] then

f(x) = \begin{cases}0 & x ≤ 0 \\ x & 0 < x < 1 \\ 1 & 1 ≤ x \end{cases}

is the Urysohn function with f(A) = 0 and f(B) = 1.

Theorem 24.5 (Tietze extension theorem). Every continuous map from a closed subspace A of a normal space X into (0, 1), [0, 1) or [0, 1] can be extended to X.

Lemma 24.6. Let X be a normal space and A ⊂ X a closed subspace. For r > 0 and any continuous map f_{0} : A → [−r, r] there exists a continuous map g: X → R such that

(24.7)  ∀x ∈ X: |g(x)| ≤ \frac{1}{3}r and ∀a ∈ A: |f_{0}(a) − g(a)| ≤ \frac{2}{3}r

Proof. Since the sets f_{0}^{−1}([−r, −\frac{1}{3}r]) ⊂ A and f_{0}^{−1}([\frac{1}{3}r, r]) ⊂ A are closed and disjoint in A they are also closed and disjoint in X. Choose (24.2) a Urysohn function g: X → [−\frac{1}{3}r, \frac{1}{3}r] such that g(x) = −\frac{1}{3}r on f_{0}^{−1}([−r, −\frac{1}{3}r]) and g(x) = \frac{1}{3}r on f_{0}^{−1}([\frac{1}{3}r, r]). From

f_{0}^{−1}([−r, r]) = \underbrace{f_{0}^{−1}([−r, −\frac{1}{3}r])}_{g=− \frac{1}{3} r} ∪ \underbrace{f_{0}^{−1}([−\frac{1}{3}r, \frac{1}{3}r])}_{|g|≤ \frac{1}{3} r} ∪ \underbrace{f_{0}^{−1}([\frac{1}{3}r, r])}_{g= \frac{1}{3} r}

we see that the second inequality of (24.7) is satisfied. □

Proof of 24.5. We shall first prove the theorem for functions from A to [0, 1] or rather to the homeomorphic interval [−1, 1] which is more convenient for reasons of notation.

Given a continuous function f: A → [−1, 1]. We have just seen (24.6) that there exists a continuous real function g_{1} : X → R such that

∀x ∈ X: |g_{1}(x)| ≤ \frac{1}{3} and ∀a ∈ A: |f(a) − g_{1}(a)| ≤ \frac{2}{3}

Now Lemma 24.6 applied to the function f − g_{1} on A says that there exists a continuous real function g_{2} : X → R such that

∀x ∈ X: |g_{2}(x)| ≤ \frac{1}{3} \frac{2}{3} and ∀a ∈ A: |f(a) − (g_{1}(a) + g_{2}(a))| ≤ (\frac{2}{3})^{2}

Proceeding this way we recursively (2.8) define a sequence g_{1}, g_{2}, ⋯ of continuous real functions on X such that

(24.8)  ∀x ∈ X: |g_{n}(x)| ≤ \frac{1}{3} (\frac{2}{3})^{n−1} and ∀a ∈ A: |f(a) − ∑_{i=1}^{n} g_{i}(a)| ≤ (\frac{2}{3})^{n}

By the first inequality in (24.8), the series ∑_{n=1}^{∞} g_{n}(x) converges uniformly and the sum function F = ∑_{n=1}^{∞} g_{n} is continuous by the uniform limit theorem (15.16). By the first inequality in (24.8),

|F (x)| ≤ \frac{1}{3} ∑_{n=1}^{∞} (\frac{2}{3})^{n−1} = \frac{1}{3} · 3 = 1,

and by the second inequality in (24.8), F (a) = f(a) for all a ∈ A.

Assume now that f: A → (−1, 1) maps A into the open interval between −1 and 1. We know that we can extend f to a continuous function F_{1} : X → [−1, 1] into the closed interval between −1 and 1. We want to modify F_{1} so that it does not take the values −1 and 1 and stays the same on A. The closed sets F_{1}^{−1}({±1}) and A are disjoint. There exists (24.2) a Urysohn function U: X → [0, 1] such that U = 0 on F_{1}^{−1}({±1}) and U = 1 on A. Then F = U · F_{1} is an extension of f that maps X into (−1, 1).

A similar procedure applies to functions f: A → [−1, 1) into the half-open interval. □

Proof of Corollary 24.1. The Urysohn lemma (24.2) says that (1) ⟹ (2) and the converse is clear. The Tietze extension theorem (24.5) says that (1) ⟹ (3). Only eg (3) ⟹ (2) remains. Assume that X is a T_{1}-space with property (3). Let A, B be two disjoint closed subsets. The function f: A ∪ B → [0, 1] given by f(A) = 0 and f(B) = 1 is continuous (13.5.(6)). Let f: X → [0, 1] be a continuous extension of f. Then f is a Urysohn function for A and B. □

Many familiar classes of topological spaces are normal.

Theorem 24.9. Compact Hausdorff spaces are normal.

Proof. See 18.7.(2). □

In fact, every regular Lindelöf space is normal [8, Ex 32.4] [5, 3.8.2].

Theorem 24.10. Metrizable spaces are normal.

Proof. Let X be a metric space with metric d and let A, B be disjoint closed sets. The continuous function f: X → [0, 1] given by

f(x) = \frac{d(x, A)}{d(x, A) + d(x, B)}

is a Urysohn function with f(A) = {0} and f(B) = {1}. This shows that X is normal (24.1). □

Theorem 24.11. [5, Problem 1.7.4]. Linearly ordered spaces are normal.

Proof. We shall only prove the special case that every well-ordered space is normal. The half-open intervals (a, b], a < b, are (closed and) open (12.2.(5)). Let A and B be two disjoint closed subsets and let a_{0} denote the smallest element of X. Suppose that neither A nor B contain a_{0}. For any point a ∈ A there exists a point x_{a} < a such that (x_{a}, a] is disjoint from B. Similarly, for any point b ∈ B there exists a point x_{b} < b such that (x_{b}, b] is disjoint from A. The proof now proceeds as the proof (23.5) for normality of R_{ℓ}. Suppose next that a_{0} ∈ A ∪ B, say a_{0} ∈ A. The one-point set {a_{0}} = [a_{0}, a_{0}^{+}) is open and closed (as X is Hausdorff). By the above, we can find disjoint open sets U, V such that A − {a_{0}} ⊂ U and B ⊂ V. Then A ⊂ U ∪ {a_{0}} and B ⊂ V − {a_{0}} where the open sets U ∪ {a_{0}} and V − {a_{0}} are disjoint. □

In particular, the well-ordered spaces S_{Ω} and \overline{S_{Ω}} are normal (the latter space is even compact Hausdorff).

25. Second countable regular spaces and the Urysohn metrization theorem

We investigate closer the class of regular spaces. We start with an embedding theorem that is used in other contexts as well.

25.1. An embedding theorem. We discuss embeddings into product spaces.

Definition 25.2. A set {f_{j}: X → Y_{j} | j ∈ J} of continuous functions is said to separate points and closed sets if for any point x ∈ X and any closed subset C ⊂ X we have

x ∉ C ⟹ ∃j ∈ J: f_{j}(x) ∉ \overline{f_{j}(C)}

Lemma 25.3. Let f: X → Y be a map that separates points and closed sets. Assume also that f is injective (eg that X is T_{1}). Then f is an embedding.

Proof. If X is T_{1}, points are closed so that f separates points, ie f is injective. For any point x ∈ X and any closed set C ⊂ X we have that

f(x) ∈ \overline{f(C)} ⟹^{f separates} x ∈ C ⟹ f(x) ∈ f(C)

so we get that f(X) ∩ \overline{f(C)} = f(C). But this equality says that f(C) is closed in f(X). Hence the bijective continuous map f: X → f(X) is closed, so it is a homeomorphism. □

Theorem 25.4 (Diagonal embedding theorem). Let {f_{j}: X → Y_{j} | ȷ ∈ J} be a family of continuous functions that separates points and closed sets. Assume that X is T_{1} or that at least one of the functions f_{j} is injective. Then the diagonal map f = (f_{j}): X → ∏_{j∈J} Y_{j} is an embedding.

Proof. Also the map f separates points and closed sets because

f(x) ∈ \overline{f(C)} ⟺ f(x) ∈ \overline{∏ f_{j}(C)} ⟺^{13.16} f(x) ∈ ∏ \overline{f_{j}(C)}

⟺ ∀j ∈ J: f_{j}(x) ∈ \overline{f_{j}(C)} ⟹^{(f_{j}) separates} x ∈ C

for all points x ∈ X and all closed subsets C ⊂ X. The theorem now follows from 25.3. □

In particular, if one of the functions f_{j} is injective and separates points and closed sets then f = (f_{j}) is an embedding. For instance, the graph X → X × Y : x → (x, g(x)) is an embedding for any continuous map g: X → Y.

25.5. A universal second countable regular space. We say that a space X is universal for some property if X has this property and any space that has this property embeds into X.

Theorem 25.6 (Urysohn metrization theorem). The following conditions are equivalent for a second countable space X:

(1) X is regular

(2) X is normal

(3) X is homeomorphic to a subspace of [0, 1]^{ω}

(4) X is metrizable The Hilbert cube [0, 1]^{ω} is a universal second countable metrizable (or normal or regular) space.

Proof. (1) ⟹ (2): Let X be a regular space with a countable basis B. We claim that X is normal. Let A and B be disjoint closed sets in X. By regularity (23.2.(1)), each point a ∈ A ⊂ X − B has a basis neighborhood U_{a} ∈ B such that a ∈ U_{a} ⊂ \overline{U}_{a} ⊂ X − B. Let U_{1}, U_{2}, … be the elements in the image of the map A → B: a ↦ U_{a}. Then

A ⊂ ⋃_{n=1}^{∞} U_{n} and \overline{U}_{n} ∩ B = ∅ for n = 1, 2 …

Similarly, there is a sequence V_{1}, V_{2}, … of basis open sets such that

B ⊂ ⋃_{n=1}^{∞} V_{n} and \overline{V}_{n} ∩ A = ∅ for n = 1, 2 …

The open sets ⋃_{n=1}^{∞} U_{n} and ⋃_{n=1}^{∞} V_{n} may not be disjoint. Consider instead the open sets

U′_{1} = U_{1} − \overline{V}_{1}  V′_{1} = V_{1} − \overline{U}_{1}
U′_{2} = U_{2} − (\overline{V}_{1} ∪ \overline{V}_{2})  V′_{2} = V_{2} − (\overline{U}_{1} ∪ \overline{U}_{2})
⋮  ⋮
U′_{n} = U_{n} − (\overline{V_{1}} ∪ ⋯ ∪ \overline{V}_{n})  V′_{n} = V_{n} − (\overline{U_{1}} ∪ ⋯ ∪ \overline{U}_{n})

Even though we have removed part of U_{n} we have removed no points from A from U_{n}, and we have removed no pints from B from V_{n}. Therefore the open sets U′_{n} still cover A and the open sets V′_{n} still cover B:

A ⊂ ⋃_{n=1}^{∞} U′_{n} and B ⊂ ⋃_{n=1}^{∞} V′_{n}

and in fact these sets are disjoint:

⋃_{n=1}^{∞} U′_{n} ∩ ⋃_{n=1}^{∞} V′_{n} = ( ⋃_{m≤n} U′_{m} ∩ V′_{n}) ∪ ( ⋃_{m>n} U′_{m} ∩ V′_{n}) = ∅

because U′_{m} ⊂ U_{m} does not intersect V′_{n} ⊂ X − U_{m} if m ≤ n and U′_{m} ⊂ X − V_{n} does not intersect V′_{n} ⊂ V_{n} if m > n. (Make a drawing of U_{1}, U_{2} and V_{1}, V_{2}.)

(2) ⟹ (3): Let X be a normal space with a countable basis B. We show that there is a countable set {f_{UV} } of continuous functions X → [0, 1] that separate points and closed sets. Namely, for each pair U, V of basis open sets U, V ∈ B such that \overline{U} ⊂ V, choose a Urysohn function f_{UV} : X → [0, 1] (24.2) such that f_{UV} (\overline{U}) = 0 and f_{UV} (X − V) = 1. Then \{f_{UV} \} separates points and closed sets: For any closed subset C and any point x ∉ C, or x ∈ X − C, there are (23.2) basis open sets U, V such that x ∈ U ⊂ \overline{U} ⊂ V ⊂ X − C (choose V first). Then f_{UV} (x) = 0 and f_{UV} (C) = 1 so that f_{UV} (x) ∉ \overline{f_{UV} (C)}. According to the Diagonal embedding theorem (25.4) there is an embedding X → [0, 1]^{ω} with the f_{UV} as coordinate functions.

(3) ⟹ (4): [0, 1]^{ω} is metrizable and any subspace of a metrizable space is metrizable. (15.5).

(4) ⟹ (1): Any metrizable space is regular, even normal (24.10). □

The point is that in a second countable regular hence normal space there is a countable set of (Urysohn) functions that separate points and closed sets. Therefore such a space embeds in the Hilbert cube.

We have (25.6, 22.2) the following identities

\begin{cases}2nd countable \\ regular \end{cases}\} = \begin{cases}2nd countable \\ normal \end{cases}\} = \begin{cases}2nd countable \\ metrizable \end{cases}\}

= \begin{cases}Countable dense subset \\ metrizable \end{cases}\} = \begin{cases}Lindelöf \\ metrizable \end{cases}\} = \begin{cases}Subspace of \\ Hilbert cube \end{cases}\}

between classes of topological spaces.

26. Completely regular spaces and the Stone–Čech compactification

There is no version of the Tietze extension theorem (24.5) for regular spaces, ie it is in general not true that continuous functions separate closed sets and points in regular spaces. Instead we have a new class of spaces where this is true.

Definition 26.1. A space X is completely regular if points are closed in X and for any closed subset C of X and any point x ∉ C there exists a continuous function f: X → [0, 1] such that f(x) = 0 and f(C) = 1.

Clearly normal ⟹ completely regular ⟹ regular ⟹ Hausdorff ⟹ T_{1}

and none of these arrows reverse (23.6, [8, Ex 33.11]).

In a completely regular space there are enough continuous functions X → [0, 1] to separate points and closed sets.

It is easy to see that any subspace of a completely regular space is completely regular [8, 33.2].

Theorem 26.2. The following conditions are equivalent for a topological space X: (1) X is completely regular

(2) X is homeomorphic to a subspace of [0, 1]^{J} for some set J

(3) X is homeomorphic to a subspace of a compact Hausdorff space

Proof. (1) ⟹ (2): If X is completely regular then the set C(X) of continuous maps X → [0, 1] separates points and closed sets. The evaluation map

Δ: X → ∏_{j∈C(X)}[0, 1], π_{j}(Δ(x)) = j(x), j ∈ C(X), x ∈ X,

is therefore an embedding (25.4).

(2) ⟹ (3): [0, 1]^{J} is compact Hausdorff by the Tychonoff theorem (18.16).

(3) ⟹ (1): A compact Hausdorff space is normal (24.9), hence completely regular and subspaces of completely regular spaces are completely regular. □

Closed subspaces of compact Hausdorff spaces are compact Hausdorff (18.7), open subspaces are locally compact Hausdorff (21.6), and arbitrary subspaces are completely regular (26.2).

Corollary 26.3. (Cf 12.19, 23.3) Any subspace of a completely regular space is completely regular. Any product of completely regular spaces is completely regular.

Proof. The first part is easily proved [8, 33.2] (and we already used it above). The second part follows from (26.2) because (13.12) the product of embeddings is an embedding. □

Corollary 26.4. Any locally compact Hausdorff space is completely regular.

Proof. Locally compact Hausdorff spaces are open subspaces of compact spaces (21.6). Completely regular spaces are subspaces of compact spaces (26.2). □

Example 26.5 (A normal space with a non-normal subspace). Take any completely regular but not normal space (for instance (23.6) R_{ℓ}×R_{ℓ}) and embed it into [0, 1]^{J} for some set J (26.2). Then you have an example of a normal (even compact Hausdorff) space with a non-normal subspace.

26.6. The Stone–Čech construction. For any topological space X let C(X) denote the set of continuous maps j: X → I of X to the unit interval I = [0, 1] and let

Δ: X → ∏_{j∈C(X)} I = map(C(X), I)

be the continuous evaluation map given by Δ(x)(j) = j(x) or π_{j}Δ = j for all j ∈ C(X). (The space map(C(X), I) is a kind of double-dual of X). This construction is natural: For any continuous map f: X → Y of X into a space Y, there is an induced map C(X) ← C(Y): f^{∗} of sets and yet an induced continuous map (13.15)

(26.7)  map(C(X), I) = ∏_{j∈C(X)} I ⟶^{f^{∗∗}} ∏_{k∈C(Y)} I = map(C(Y), I)

such that the diagram

(26.8)  \xymatrix{X \ar[rr]^{f} \ar[d]_{Δ_{X}} &  & Y \ar[d]^{Δ_{Y}} \\ ∏_{j∈C(X)} I \ar[rr]_{f^{∗∗}} \ar[dr]_{π_{f^{∗}k}=π_{kf}} &  & ∏_{k∈C(Y)} I \ar[dl]^{π_{k}} \\  & I}

commutes: The lower triangle commutes by the definition of the map f^{∗∗} (13.15) and then the upper square commutes because π_{k}f^{∗∗}Δ_{X} = π_{kf}Δ_{X} = kf = π_{k}Δ_{Y} f. Put βX = \overline{Δ(X)} and define Δ: X → βX to be the corestriction of Δ: X → ∏_{j∈C(X)} I to βX. Then βX is a compact Hausdorff space (it is a closed subspace of the compact (18.16) Hausdorff space ∏ I) and Δ is, by design, a continuous map with a dense image ΔX in βX. This construction is natural: For any continuous map f: X → Y, the induced map f^{∗∗} (26.7) takes βX into βY for

f^{∗∗}(βX) = f^{∗∗}(\overline{Δ_{X}X}) ⊂ \overline{f^{∗∗}Δ_{X}X} =^{(26.8)} \overline{Δ_{Y} f(X)} ⊂ \overline{Δ_{Y} Y} = βY

and thus we obtain from (26.8) a new commutative diagram of continuous maps

(26.9)  \xymatrix{X \ar[r]^{f} \ar[d]_{Δ_{X}} & Y \ar[d]^{Δ_{Y}} \\ βX \ar[r]_{βf} & βY}

where βf: βX → βY is the corestriction to βY of the restriction of f^{∗∗} to βX. (This makes β into a functor with a natural transformation from the identity functor from the category of topological spaces to the category of compact Hausdorff spaces.)

The next result says that the map X → βX is the universal map from X to a compact Hausdorff space.

Theorem 26.10. Let X be a topological space.

(1) The map X → βX is a continuous map of X into a compact Hausdorff space.

(2) For any continuous map f: X → Y of X into a compact Hausdorff space Y there exists a unique continuous map \overline{f}: βX → Y such that the diagram

(26.11)  \xymatrix{X \ar[dr] \ar[rr]^{f} &  & Y \\  & βX \ar[ur]_{\overline{f}}}

commutes. (We say that f: X → Y factors uniquely through βX.)

(3) Suppose X → αX is a map of X to a compact Hausdorff space αX such that any map of X to a compact Hausdorff space factors uniquely through αX. Then there exists a homeomorphism αX → βX such that

\xymatrix{ & X \ar[dl] \ar[dr] \\ αX \ar[rr]^{≃} &  & βX}

commutes.

(4) If X is completely regular then X → βX is an embedding. If X is compact Hausdorff then X → βX is a homeomorphism.

Proof. (4): Since the corestriction of an embedding is an embedding (13.12) and X → ∏ I is an embedding when X is completely regular (26.2), also X → βX is an embedding. If X is compact Hausdorff, X is normal (24.9), hence completely regular, so we have just seen that Δ: X → βX is an embedding. The image of this embedding is closed (18.8.(1)) and dense. Thus the embedding is bijective so it is a homeomorphism.

(1) and (2): If Y is compact Hausdorff then by item (4) Δ_{Y} is a homeomorphism in diagram (26.9) and hence \overline{f} = Δ_{Y}^{−1} ∘ βf is a possibility in (26.11). It is the only possibility [8, Ex 18.13], for the image of X is dense in βX.

(3) Let X → αX be a map to a compact Hausdorff space satisfying the above universal property. Then there exist continuous maps \xymatrix{αX \ar[r] & βX \ar[l]} that, by uniqueness, are inverse to each other. (All universal constructions are essentially unique.) □

If X is completely regular then X → βX is a compactification (26.10.(4)) and it is called the Stone–Čech compactification of X. Conversely, if X has a compactification then X is homeomorphic to a subspace of a compact space and therefore (26.2) completely regular. We state these observations in

Lemma 26.12. X has a compactification (21.2) if and only if X is completely regular.

The Stone–Čech compactification βX of a completely regular space X is the maximal compactification in the sense that for any other compactification X → cX there is a closed quotient map βX → cX map such that

\xymatrix{ & X \ar[dl] \ar[dr] \\ βX \ar[rr] &  & cX}

commutes. The map βX → cX is closed by 18.8 and surjective because its image is closed and dense.

The Alexandroff compactification ωX of a noncompact locally compact Hausdorff space X is the minimal compactification in the sense that ωX = cX/(cX − X) for any other compactification X → cX.

Indeed we saw in 21.3 that there is a closed quotient map cX → ωX, taking cX − X to ω, such that

\xymatrix{ & X \ar[dl] \ar[dr] \\ cX \ar[rr] &  & ωX}

commutes. (What is the minimal compactification of a compact Hausdorff space?)

For instance, there are quotient maps

\xymatrix{ & [0, 1] \ar[dr] \\ βR \ar@{-->}[ur] \ar[dr] &  & ωR = S^{1} \\  & C \ar[ur]}

of compactifications of R. Here

C = \{0\} × [−1, 1] ∪ \{(x, sin \frac{1}{x}) | 0 < x ≤ π^{−1}\} ∪ L

is the Warsaw circle, a compactification of R with remainder C −R = [−1, 1], which is obtained by closing up the closed topologist’s sine curve \overline{S} by (a piece-wise linear) arc from (0, 0) to (π^{−1}, 0). In particular, C/[−1, 1] = S^{1}. More generally, there are quotient maps βR^{n} → [0, 1]^{n} → ωR^{n} = S^{n} of compactifications of euclidean n-space R^{n}.

Investigations of the Stone–Čech compactification of the integers βZ raise several issues of a very fundamental nature [12, 13, 10] whose answers depend on your chosen model of set theory.

27. Manifolds

Definition 27.1. A manifold is a locally euclidean second countable Hausdorff space.

Manifolds are locally compact and locally path connected since they are locally euclidean. All locally compact Hausdorff spaces are (completely) regular. Thus manifolds are second countable regular spaces and hence they are normal, metrizable, Lindelöfand they have countable dense subsets.

The line with two zeroes is an example of a locally euclidean space that is not Hausdorff. The long line [8, Ex 24.12] is a locally euclidean space that is not 2nd countable.

The n–sphere S^{n} and the real projective n–space RP^{n} (14.15) are examples of manifolds. S^{n} lies embedded in R^{n+1} from birth but what about RP^{n}? Does RP^{n} embed in some euclidean space?

Theorem 27.2 (Embeddings of compact manifolds). Any compact manifold embeds in R^{N} for some N.

Proof. We may assume that M is connected. Let n be the dimension of M (we assume that this is well-defined). Let B = {x ∈ R^{n} | |x| ≤ 1} be the unit ball in R^{n}. Cover M by finitely many closed sets B_{1}, …, B_{k} such that each B_{i} is homeomorphic to B and such that the interiors of the B_{i} cover M. Let f_{i}: M → S^{n} be the continuous map obtained by collapsing the complement to the interior of B_{i} to a point. The map f = (f_{1}, …, f_{k}): M → S^{n} × ⋯ × S^{n} is injective: If x lies in the interior of B_{i} then f_{i}(x) ≠ f_{i}(y) for all y ≠ x. Since M is compact and (S^{n})^{k} Hausdorff, f is an embedding. Finally, each sphere S^{n} embeds into R^{n+1} so that (S^{n})^{k} embeds into (R^{n+1})^{k}. □

In particular RP^{2} (14.15) embeds into some euclidean space. Which one?

There is up to homeomorphism just one compact manifold of dimension 1, the circle. Compact manifolds of dimension 2 are described by a number, the genus, and orientability. In particular, any simply connected 2-dimensional compact manifold is homeomorphic to S^{2}. We do not know (December 2003) if there are any simply connected compact 3-dimensional manifolds besides S^{3} (Poincaré conjecture). Classification of 4-dimensional compact manifolds is logically impossible.

Any manifold admits a partition of unity. We shall prove the existence in case of compact manifolds.

Theorem 27.3 (Partition of unity). Let X = U_{1} ∪ ⋯ ∪ U_{k} be a finite open covering of the normal space X. The there exist continuous functions ϕ_{i}: X → [0, 1], 1 ≤ i ≤ k, such that

(1) \overline{\{ϕ_{i} > 0\}} ⊂ U_{i}

(2) ∑_{i=1}^{k} ϕ_{i}(x) = 1 for all x ∈ X.

Proof. We show first that there is an open covering \{V_{i}\} of X such that \overline{V}_{i} ⊂ U_{i} (we shrink the sets of the covering). Since the closed set X − (U_{2} ∪ ⋯ ∪ U_{k}) is contained in the open set U_{1} there is an open set V_{1} such that

X − (U_{2} ∪ ⋯ ∪ U_{k}) ⊂ V_{1} ⊂ \overline{V}_{1} ⊂ U_{1}

by normality (23.2). Now V_{1} ∪ U_{2} ∪ ⋯ ∪ U_{k} is an open covering of X. Apply this procedure once again to find an open set V_{2} such that \overline{V}_{2} ⊂ U_{2} and V_{1} ∪ V_{2} ∪ U_{3} ∪ ⋯ ∪ U_{k} is still an open covering of X. After finitely many steps we have an open covering \{V_{i}\} such that \overline{V}_{i} ⊂ U_{i} for all i.

Do this one more time to obtain an open covering \{W_{i}\} such that W_{i} ⊂ \overline{W}_{i} ⊂ V_{i} ⊂ \overline{V}_{i} ⊂ U_{i} for all i. Now choose a Urysohn function (24.2) ψ_{i}: X → [0, 1] such that ψ_{i}(\overline{W}_{i}) = 1 and ψ_{i}(X −V_{i}) = 0. Then \overline{\{x | ψ_{i}(x) > 0\}} ⊂ \overline{V}_{i} ⊂ U_{i} and ψ(x) = ∑ ψ_{i}(x) > 1 for any x ∈ X since x ∈ W_{i} for some i. Hence ϕ_{i} = \frac{ψ_{i}}{ψ} is a well-defined continuous function on M taking values in the unit interval such

that ∑_{i=1}^{k} ϕ_{i}(x) = ∑_{i=1}^{k} \frac{ψ_{i}(x)}{ψ(x)} = \frac{1}{ψ(x)} ∑_{i=1}^{k} ψ_{i}(x) = \frac{ψ(x)}{ψ(x)} = 1. □

28. Relations between topological spaces

\xymatrix{ &  & sequentially compact \ar@{=>}[d]^{+metrizable} \\ countable dense subset \ar@{=>}[d]^{+metrizable} &  & limit point compact \ar@{=>}[d]^{+metrizable} \ar@{=>}[u]^{+1st countable+T_{1}} \\ 2nd countable \ar@{=>}[d] \ar@{=>}[u]^{Thm 30.3} \ar@{=>}[r]_{Thm 30.3} & Lindelöf \ar@{=>}[l]_{+metrizable} & compact \ar@{=>}[r] \ar@{=>}[l] \ar@{=>}[u]^{Thm 28.1} & locally compact \\ 1st countable \ar@{=>}[r]_{Thm 30.1} & Sequence lma & compact Hausdorff \ar@{=>}[u] \ar@{=>}[d] \ar@{=>}[r] & locally compact Hausdorff \ar@{=>}[u] \ar@{=>}[d]_{Ex 22.7} \\ metrizable \ar@{=>}[u] \ar@{=>}[r]_{Ex 32.7} & completely normal \ar@{=>}[r] & normal \ar@{=>}[r]_{Thm 33.1} & completely regular \ar@{=>}[d] & top grp \ar@{=>}[l]^{Ex 33.10} \\ regular 2nd countable \ar@{=>}[u]^{Thm 34.1} & well-ordered topology \ar@{=>}[r] \ar@{=>}[d]_{Ex 23.5} & order topology \ar@{=>}[u]^{Thm 32.4} & regular \ar@{=>}[d] \\ manifold \ar@{=>}[u]^{Ex 36.1} \ar@{=>}[ddr] & totally disconnected & linear continuum \ar@{=>}[u] \ar@{=>}[d]_{Thm 24.1} & Hausdorff \ar@{=>}[d] \\  & locally connected & connected & T_{1} \\  & locally path connected \ar@{=>}[u] & path connected \ar@{=>}[u]^{p. 155}}

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