Eksamen juni 2004

Matematik 3GT

Books, notes, calculators, and computers are allowed at this three hour written exam. You may write your answers in pencil.

Problem 1

Let f: X → Y be a map between two topological spaces. Show that the following two statements are equivalent:

(1) f is continuous and closed

(2) \overline{f(A)} = f(\overline{A}) for all A ⊂ X.

Problem 2 Let f: K → X be a continuous map of a compact space K into a locally pathconnected Hausdorff space X. Show that the path-components of X − f(K) are open.

Problem 3

Let X be a normal space and βX its Stone–Čech compactification. We consider X as a subset of βX. Let y be a point in the remainder βX − X. Assume that y is the limit of a sequence of points from X. We will show that this assumption leads to a contradiction.

(1) Show that y is the limit of a sequence of points x_{n} ∈ X, n ∈ Z_{+}, all of whose points are distinct, ie such that x_{n} ≠ x_{m} for n ≠ m. [Hint: If (x_{n}) converges to y, where x_{n} is in X and y in βX − X, consider the subsequence (x_{n_{k}}) where

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}

is defined recursively.]

(2) Let A = {x_{1}, x_{3}, x_{5}, …} be the set of odd and B = {x_{2}, x_{4}, x_{6}, …} the set of even elements of the sequence. Show that \overline{A} = A ∪ \{y\} and \overline{B} = B ∪ \{y\}. What is \overline{A} ∩ \overline{B}? (Closures are taken in βX.)

(3) Show that A and B are are disjoint closed subsets of X and that there exists a continuous function f: X → [0, 1] such that f(A) = 0 and f(B) = 1.

(4) Show that there exists a continuous function \overline{f}: βX → [0, 1] such that f(A) = 0 and f(B) = 1. What does that tell you about \overline{A} ∩ \overline{B}?

(5) Explain why we now have a contradiction and conclude that no point of

βX − X is the limit of a sequence of points from X. Assume now that X is normal and noncompact.

(6) Explain why βX − X is nonempty.

(7) Explain why βX does not satisfy the conclusion of the Sequence lemma.

Conclude that βX is not first countable and not metrizable.

(THE END)