Purpose of Metrizability What is an example of when metrizability is used to prove a result? Or is metrizability of a more philosophical nature?
 A: 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.

