Master’s Comprehensive Exam—Topology—January 1996

1. Prove that every metric space is normal.

2. X is a topological space. There exists a basis B for the topology on X such that any two distinct members of B are disjoint. What’s more, X is Hausdorff. Prove that X has the discrete topology.

3. (a) Let X be a Hausdorff space. Let K_{1}, K_{2} be compact subsets of X. Prove that K_{1} ∩ K_{2} is compact.

(b) Give a counterexample when X is not Hausdorff. Hint: let T be a space with two points in the indiscrete topology. Consider ℝ × T.

4. Let {X_{α}} be a family of topological spaces, with α ranging through an index set J. For all α, let Y_{α} be an open subset of X_{α}. Consider the space X whose points are J-tuples x = (x_{α})_{α∈J} such that

(1) x_{α} ∈ X_{α} for every α ∈ J, and

(2) x_{α} ∈ Y_{α} for all but finitely many α ∈ J. (Which α are left out depends on x.) We give X a topology by taking as a basis the sets

∏_{α∈J} U_{α},

where U_{α} is open in X_{α} for all α, and U_{α} = Y_{α} for almost all α. With this topology, we call X the restricted topological product of the X_{α} with respect to the Y_{α}.

(a) Fix a finite set S ⊂ J. Let X_{S} be the set of all x ∈ X such that x_{α} ∈ Y_{α} whenever α ∉ S. Explain why X_{S} is open in X.

(b) Notice that, as a set,

(4.1)  X_{S} = ∏_{α∈S} X_{α} × ∏_{α∉S} Y_{α}.

So it is natural to give X_{S} the product topology coming from (4.1). On the other hand, since X_{S} ⊆ X, it has a subspace topology coming from X. Prove that this subspace topology and that product topology are the same.

(c) Assume in addition that each X_{α} is locally compact and that each Y_{α} is compact. Prove that X is locally compact. Hint: use the X_{S}.

Typeset by AMS-TEX