E-2 - Classical Metrization Theorems

Abstract

A topological space is said to be metrizable if there is a metric d on X such that the topology induced by d is ▪. A metrization theorem gives (necessary and) sufficient conditions for a space to be metrizable. According to the Alexandroff–Urysohn theorem, a topological space is metrizable if and only if (iff) it has a regular development and a regular space with a countable base is metrizable as per Urysohn. Bing's Theorem states that a topological space is metrizable iff it is regular and has a σ -discrete base. The Alexandroff–Urysohn Theorem clearly establishes the concept of a development as fundamental in metrization theory. Urysohn's Theorem is not easily derived from the Alexandroff–Urysohn Theorem, which gives rise to the famous metrization problem of finding a necessary and sufficient condition for metrizability that gives Urysohn's result as an easy corollary. Bing's Theorem introduces into topology a new and important separation axiom called collection wise normality. A.H. Stone's important result which says that every metric space is paracompact plays a key role in the Nagata–Smirnov–Bing solution of the metrization problem.