CHAPTER VI - METRIZABLE SPACES AND RELATED TOPICS

Abstract

This chapter discusses the theories on metrizable spaces, the properties of metrizable spaces, and the concept of metric space. There are various theorems that have been established in the field of metrization theory but still they do not seem to be exhausted. A topological space R is called metrizable if it is homeomorphic with a metric space. In metrization theory, the classical theorem known as Alexandroff–Urysohn's metrization theorem is still fundamental. The chapter discusses Nagata–Smimov's metrization theorem and Bing's metrization theorem. The chapter discusses the problem of topologically imbedding a given metrizable space in a concrete space. A topological space R is generally said to be topologically imbedded in a topological space S if R is homeomorphic with a subspace of S. The chapter also discusses metric spaces, compact spaces, and other important spaces as special cases.