What is a good application of Urysohn's Theorem? 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).
 A: 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.
A: 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.
The theorem states that if $(X,\tau)$ is a locally compact Hausdorff space, then for every positive linear functional $\Lambda : C_{0}\to \mathbb{R}$ there exists a Borel measure $\mu$ so that for all $f\in C_{0}$:
$$\Lambda (f) = \int_{X} f d \mu$$
Where $C_{0}$ is the collection of all compactly supported real-valued continuous functions on $X$. Conversely, every functional defined as above for a given measure is linear and positive, so this theorem gives one-to-one correspondence with Borel measures and linear positive functionals.
The proof relies explicitly on Urysohn's lemma and partitions of unity that are obtained aswell by using this lemma. Note that a locally compact Hausdorff space is $T_{3}$.
You can find the detailed proof if you're interested from: Rudin W., Real and Complex analysis, 1970, Theorem 2.14
