This chapter discusses the Alexandroff-Urysohn metrization theorem. Given a metrizable space (X, T), there are two standard methods of constructing a metric on X compatible with the topology T. One method is to start with a σ-locally finite or σ-discrete base for X, use of Urysohn's Lemma to obtain a collection of continuous, real-valued functions on X, and then use these functions to construct the metric. The other method is to begin with a regular development for X, obtain a distance function on X, and then construct a metric using this distance function. Both of these constructions are basic tools in general topology. The chapter discusses the applications of the first construction (obtaining a suitable family of continuous pseudo-metrics).
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