Master’s Comprehensive Exam

General Topology, August 1997

1. (10pts) Let X be the set {0, 1, 2, 3, 4} and T be a topology of X given by

T = {X, ∅, {0}, {2, 3}, {0, 2, 3}, {1, 2, 3, 4}}.

(i) Find all closed subsets of X.

(ii) Find interior points and exterior points of A = {1, 2, 3}.

(iii) Find the closure of A. Is (X, T) a Hausdorff space ?

2. (20pts) Prove that a completely regular space is regular.

Recall: A topological space X is completely regular if it satisfies the following. For a closed subset F of X and p ∈ X \ F, there exists a continuous function f : X → [0, 1] such that f(p) = 0 and f(F) = 1.

3. (20pts) Let f : (X, T_{X}) → (Y, T_{Y}) be a one-to-one continuous function. If (X, T_{X}) is compact and (Y, T_{Y}) is Hausdorff, then (X, T_{X}) is homeomorphic to (f(X), T_{Y} |_{f(X)}) ( T_{Y} |_{f(X)} is the induced topology of f(X)).

4. (20pts) If A is a totally bounded subset of a metric space X, then every sequence {a_{n}} in A contains a Cauchy subsequence.

5. (20pts) Let f : [0, 1] → [0, 1] be a continuous function. Show that there exists a point a ∈ [0, 1] such that f(a) = a.

6. (10pts) Show that every regular Lindelöf space is normal.