F-10 - Topological Characterizations of Spaces

Abstract

Despite over a century of attention, there are surprisingly few topological spaces that can be given succinct elementary characterizations in purely topological terms. This chapter lists some of the more prominent spaces, restricted to the separable metric spaces, which are characterized by Urysohn's Metrization Theorem as the regular Hausdorff spaces satisfying the second axiom of countability. It begins with the simplest and most widely applied Cantor's middle third set which states that a non-void topological space X is homeomorphic with the Cantor set C provided that it is compact, metrizable, zero-dimensional, and perfect. It then reviews other subsets of the real line. If X is a connected topological space, a subset S ⊂X separates X provided that X-S is equal to A⋃B, where A and B are open, disjoint, and non-empty. X is non-degenerate provided that it has more than one point. A component of a space is a maximal connected set. A space homeomorphic with the closed interval is frequently called an arc; if the end points, the ones that do not separate the arc, are deleted, the result, homeomorphic with the real numbers, is often termed an open arc.