Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.
What is an example of a Hausdorff second-countable regular space where it is difficult to prove metrizability without using Urysohn's Theorem?
For example, the theorem implies that a (second-countable) manifold is metrizable. However, this result can be proven without using Urysohn's Theorem by showing directly that every such manifold embeds in $\mathbb{R}^{\infty}$ (using partitions of unity).
Best Answer
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.