A metrization theorem

Abstract

A. characterization of metrizable topological spaces in terms of subtopologies is given. First, several terms are defined in order to describe the pertinent subtopologies. Then, the characterization is readily established as a result of a metrization theorem due to Bing [l ] and a metrization theorem due to Ceder [2]. Definition 1. Let (X, 3) be a topological space having the property that the intersection of an arbitrary number of open sets is open. Then (X, 3) is called a fundamental topological space. Definition 2. Let (X, 3) be a topological space with a base P— {Bj-. jCZ-J} such that for each x£(~) {5,-: i£J, Ian arbitrary subset of /} there exists BXCZP where xCZBxCZV\{Bi: *£/}. Then, P is called a fundamental base for 3. Remark. If (X, 3) has a fundamental base for 3, it is a fundamental space. Definition 3. Let (X, 3) be a topological space. Let Pi, a subcollection of 3, be a base for some topology, 3,-, on X. Consider the subcollection of 3, p~it where /3<= {X—Tt: 7\£3i and the closure is with respect to 3}. Then, Pi is a base for a topology on X, 3/3*. The topology, 3/3* is called the dual topology of 3, relative to 3. Pi is called the dual base of pt. We state two metrization theorems, without proof, to be used in the establishment of a new metrization theorem. For other proofs see [1], [2].