Chapter X. Topological Spaces

Abstract

This chapter extends considerably the framework for discussing convergence, limits, and continuity that was developed in Chapter II: topological spaces replace metric spaces.

Section 1 makes various definitions, including definitions for the terms topology, open set, closed set, continuous function, base for a topology, separable, and subspace. It introduces two general kinds of constructions useful in analysis and other fields for forming new topological spaces out of old ones—weak topologies and quotient topologies. The section gives several examples of each.

Sections 2–3 develop standard facts, mostly elementary, about how certain combinations of properties of topological spaces imply others. Examples show some limitations to such implications. Properties that are studied include Hausdorff, regular, normal, dense, compact, locally compact, Lindelöf, and $\sigma$-compact.

Section 4 discusses product topologies on arbitrary product spaces, an example of a weak topology. The main theorem, the Tychonoff Product Theorem, says that the product of compact spaces is compact.

Section 5 introduces nets, a generalization of sequences. Sequences by themselves are inadequate for detecting convergence in general topological spaces, and nets are a substitute. The use of nets in many cases provides an easier way of establishing properties of subsets of a topological space than direct arguments with open and closed sets.

Section 6 elaborates on quotient topologies as introduced in Section 1. Conditions under which a quotient space is Hausdorff are of particular interest.

Sections 7–8 prove and apply Urysohn's Lemma, which says that any two disjoint closed sets in a normal topological space may be separated by a real-valued continuous function. This result is fundamental to serious uses of topological spaces in analysis. One application is to showing that every separable Hausdorff regular topology arises from a metric.

Section 9 extends Ascoli's Theorem and the Stone–Weierstrass Theorem from their settings in compact metric spaces in Chapter II to the wider setting of compact Hausdorff spaces.