Metrization theorems

Abstract

The goal of this study is to prove The Urysohn Metrization Theorem. This paper represents an introduction to topological spaces with the focus on metric spaces. We provide a background in set theory and function theory first, then proceed introducing the distance function and looking at some examples of metric spaces, especially the Euclidean n-space. The overview of topological spaces in general leads us to product spaces. Our study of connectedness and separability of topological spaces paves the way to the separation by continuous functions. In conclusion, the proofs of Urysohn's Lemma and The Tietze Extension Theorem enable us to prove The Urysohn Metrization Theorem.