Appendix A - Point Set Topology

Summary

In this appendix, we present some topological ingredients used in the main body of this book. In particular, the results stated are confined mostly to the metric space set up and most of the time without the proofs.

Urysohn's Lemma and Tietze Extension Theorem

Let (X, d) be a metric space with metric d. Let A be a non-empty subset of X, and for x ∈ X, let d(x, A) = inf﹛d(x, a) : a ∈ A﹜. Then, d(x, A) is a continuous function of x. Let A and B be disjoint non-empty closed subsets of X. For x ∈ X, define

Clearly, f : X → [0, 1] is a continuous function. Note that f (a) = 0 and f (b) = 1 for every a ∈ A and b ∈ B. In particular, the disjoint non-empty closed subsets A and B of X are separated by the continuous function f. Thus, we obtain the following special case of Urysohn's lemma (refer to [96]).

Theorem A.1.1 (Urysohn's Lemma)

Let X be a metric space. Given closed non-empty disjoint subsets A and B of X, there exists a continuous function f : X → [0, 1] such that and.

The following particular consequence of Urysohn's lemma is invoked in Chapter 3.

Corollary A.1.1

Let X be a compact Hausdorff space. Let be such that g vanishes on a non-empty closed subset K of X. For any, there exists a continuous function G on X vanishing in a neighborhood of K such that

Proof Note that there exists a neighborhood V of K such that. Let U be an open subset of X such that. Apply Urysohn's lemma to U and to get such that on U and f = 1 on. Let G = gf.

We also need the following extension of Urysohn's lemma in the main body.

Theorem A.1.2 (Tietze Extension Theorem)

Any continuous real-valued function on a closed subspace of a metric space may be extended to a continuous real-valued function on the entire space.

Product Topology and and Tychonoff's Theorem

Consider an arbitrary family of topological spaces indexed by a set I. The product topology on is the topology generated by the basis

For, one may define the projection map, where. It is worth mentioning that the product topology is the smallest topology that makes all projections continuous.