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).