Metrization of Topological Spaces

Abstract

A single valued function D(x, y) is a metric for a topological space provided that for points x, y, z of the space: 1. the equality holding if and only if x = y, 2. (symmetry), 3. (triangle inequality), 4. x belongs to the closure of the set M if and only if P(x, m) (m element of M) is not bounded from 0 (preserves limit points).