Eksamen juni 2004

Solutions to the June 2004 exam

Problem 1

(1) ⟹ (2): Since f is continuous, f(A) ⊂ f(\overline{A}) ⊂ \overline{f(A)}. Since f is closed, f(\overline{A}) is closed and therefore these two inclusions imply \overline{f(A)} = f(\overline{A}).

(2) ⟹ (1): Since f(\overline{A}) ⊂ \overline{f(A)} for all A ⊂ X, f is continuous. If A is closed, then A = \overline{A} so that f(A) = f(\overline{A}) = \overline{f(A)} is also closed. This means that f is closed. Problem 2 Any continuous map of a compact space to a Hausdorff space is closed. Open subsets of locally path-connected spaces have open path-components. Problem 3 [2, Ex 38.9]

(1) Suppose that x_{n} ∈ X converges to y ∈ βX − X. We recursively define a subsequence by x_{n_{k}} by

n_{k} = \begin{cases}1 & k = 1 \\ min\{n > n_{k−1} | x_{n} ∉ \{x_{n_{1}}, …, x_{n_{k−1}}\}\} & k > 1 \end{cases}

This definition makes sense since the set we are taking the minimal element of a nonempty set. Since x_{n} converges to y, the subsequence x_{n_{k}} also converges to y. Clearly, no two points of the subsequence x_{n_{k}} are identical. We call this subsequence x_{n} again.

(2) Note that y is in the closure of A as any neighborhood of y contains a point from A. Since therefore A ⊂ A ∪ \{y\} ⊂ \overline{A}, it suffices to show that A ∪ \{y\} is closed, ie that the complement of A∪{y} is open: Let z be a point in the complement. Since z is not the limit of the sequence (x_{2n+1}) (there is just one limit point, namely y, in the Hausdorff space βX) there exists a neighborhood, U, of z containing only finitely many points from the sequence. Then U − (U ∩ (A ∪ {y})) is an open neighborhood of z disjoint from A ∪ {y}.

This shows that \overline{A} = A∪\{y\}. Similarly, \overline{B} = B∪\{y\}. Therefore the intersection \overline{A} ∩ \overline{B} = \{y\} ≠ ∅.

(3) A and B are disjoint since no two points of the sequence x_{n} are identical. The set A is closed in X since Cl_{X} A = X ∩ \overline{A} = X ∩ (A ∪ \{y\}) = A. Similarly for B, of course. By Urysohn’s characterization of normal spaces, there exists a continuous function f: X → [0, 1] such that A ⊂ f^{−1}(0) and B ⊂ f^{−1}(1).

(4) The universal property of the Stone–Čech compactification [1, §27] says that there exists a unique continuous map \overline{f} into the compact Hausdorff space [0, 1] such that the diagram

\xymatrix{X \ar[rr]^{f} \ar[dr] &  & [0, 1] \\  & βX \ar[ur]_{\overline{f}}}

commutes. Since \overline{A} ⊂ \overline{f}^{−1}(0) and \overline{B} ⊂ \overline{f}^{−1}(1), \overline{A} and \overline{B} are disjoint.

(5) ∅ ≠^{(2)} \overline{A} ∩ \overline{B} =^{(4)} ∅ ⚡

(6) X is a proper subspace of βX since βX is compact which X is not.

(7) First countable spaces satisfy the sequence lemma. If βX−X = \overline{X}−X is nonempty then βX does not satisfy the sequence lemma as we have just seen. Thus βX is not first countable, in particular not metrizable.

Here are two applications:

• βZ_{+} is compact but not sequentially compact: The sequence Z_{+} in βZ_{+} has no convergent subsequence (neither in Z_{+} nor in βZ_{+} − Z_{+}).

• βR is connected but not path-connected: βR is connected because it has a dense connected subspace, R. It is not path-connected since there can be no path between a point in R and a point outside R as any such path would also give a a sequence of points in R converging to a point outside R (the first point of the path that is not in R). To see this note that as R is locally compact, R is open in the compact Hausdorff space βR [1, 30.8]. The remainder βR − R is therefore closed. Let now p: [0, 1] → βR be a path from p(0) ∈ R to p(1) ∈ βR − R. Then p^{−1}(βR − R) is a compact subset of [0, 1] so it contains its greatest lower bound t_{0} = inf p^{−1}(βR − R). This lower bound is positive for 0 belongs to p^{−1}(R). Hence t_{n} = t_{0} − \frac{1}{n} is a sequence in p^{−1}(R) (for n ≫ 0) that converges to t_{0} ∉ p^{−1}(R). The image sequence p(t_{n}) is then a sequence in R that converges to a point p(t_{0}) outside R. But that is impossible!

References

[1] Jesper M. Møller, General topology, http://www.math.ku.dk/ moller/e03/3gt/notes/gtnotes.dvi.

[2] James R. Munkres, Topology. Second edition, Prentice-Hall Inc., Englewood Cliffs, N.J., 2000. MR 57 #4063