Purpose of Metrizability

Question

What is an example of when metrizability is used to prove a result? Or is metrizability of a more philosophical nature?

Answer 1

From a question on mathoverflow

Some comments from the link mentioned above.

A very remarkable and classical result that uses repeatedly the Urysohn's lemma (not the metrization theorem) is the proof of Riesz representation theorem in its general setting.

2.

One good use is to conclude that the unit ball of the dual of a separable Banach space, in the w*-topology, is a compact metric space.

3.

This doesn't answer your question, but it's worth noting that the proof of Urysohn's Theorem easily gives what Munkres calls the Imbedding Theorem (34.2), since you end up imbedding X into some giant RJ. This characterizes completely regular spaces as subspaces of compact Hausdorff spaces. In turn, that theorem is used to prove the Nagata-Smirnov metrization theorem, which actually classifies metric spaces. To me, that's reason enough to develop Urysohn's theorem

4.

The proof of Urysohn's metrization theorem provides you with a more or less explicit metric coming from an embedding into a product space (the metric looks similar to what I wrote in a comment to an answer) and is related to what you described as a way to circumvent Urysohn's theorem when proving metrizability of manifolds: if you can prove your space to be Hausdorff, regular and second countable then you can write down a metric for its topology.