Paracompact Hausdorff spaces are very close to metrizable spaces, and have proved quite useful in differential geometry and topology. Section 9.1 concerns with the study of these spaces. It will be seen that every metric space is paracompact Hausdorff. On the other hand, a locally metrizable paracompact Hausdorff space is metrizable. In fact, there are several theorems that guarantee the metrizability of a topological space X. In the previous chapter, we have seen such a theorem, viz., the Urysohn Metrization Theorem. In Sect. 9.2, we treat the most important one, it gives a complete characterization of metric spaces.