Eksamen januar 2005

JMM

Matematik 3GT

Books, notes, calculators, and computers are allowed at this three hour written exam. You may write your answers in pencil. An unjustified answer counts as no answer.

Problem 1

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

(1) f is continuous and open.

(2) f^{−1}(Int(B)) = Int(f^{−1}(B)) for all B ⊂ Y.

(3) f^{−1}(\overline{B}) = \overline{f^{−1}(B)} for all B ⊂ Y.

Problem 2

Let X be a topological space and A ⊂ X a subset. The exterior and the boundary of A are the subsets

Ext(A) = X − \overline{A}, Bd(A) = \overline{A} ∩ \overline{X − A}

of X.

(1) Explain why the exterior of A is open and the boundary of A is closed.

(2) Show that X = Int(A) ∪ Bd(A) ∪ Ext(A) and that these three sets are pairwise disjoint.

(3) Let u: [0, 1] → X be a continuous path in X from a point u(0) ∈ Int(A) in the interior of A to a point u(1) ∈ Ext(A) in the exterior of A. Show that there is a t ∈ [0, 1] such that u(t) is in the boundary, Bd(A), of A.

Problem 3

Consider the spaces

{0} ∪ {1/n | n ∈ Z_{+}} = A ⊂ I = [0, 1]

equipped with their standard topologies (as subspaces of R with the standard topology). Show that there does not exist a continuous map R: I × I → I × {0} ∪ A × I such that R(x × t) = x × t for all x × t in I × {0} ∪ A × I.

You may proceed as follows: Assume that the map R exists.

(1) Show that R(x_{n} × 1) = \frac{1}{2} (\frac{1}{n+1} + \frac{1}{n}) × 0 for some point x_{n} ∈ [\frac{1}{n+1}, \frac{1}{n}].

(2) Obtain a contradiction with the continuity of R.

Problem 4 Let X be a Hausdorff space equipped with an ascending sequence of closed subspaces X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n−1} ⊂ X_{n} ⊂ ⋯ ⊂ X. Assume that the topology on X is coherent with this filtration in the sense that X = ⋃_{n=0}^{∞} X_{n} and

A is closed ⟺ A ∩ X_{n} is closed for each n

holds for all subsets A of X.

Let C be a compact subset of X. Choose a point t_{n} ∈ C ∩ (X_{n} − X_{n−1}) for all n ∈ Z_{+} for which this intersection is nonempty and let T be the set of all the points t_{n}.

(1) Show that T is closed and that any subspace of T is closed. What can you say about the subspace topology on T?

(2) Show that T is finite.

(3) Conclude that C is contained in X_{N} for some N ∈ Z_{+}.

(THE END)