Urysohn metrization theorem

Remarks

A second-countable regular space is normal, so Urysohn's lemma applies to it. With a countable base one picks countably many continuous functions $f_n \colon X \to [0, 1]$ that separate points from closed sets. The map $x \mapsto (f_n(x))_n$ is then an embedding of $X$ into the Hilbert cube $[0,1]^{\N}$, which is metrizable. For example, every subspace of $\R^n$ satisfies the hypotheses, and so does every compact Hausdorff space with a countable base. Second countability cannot be dropped: the Sorgenfrey line is regular and Hausdorff but not metrizable. Regularity cannot be dropped either: there are second-countable Hausdorff spaces that are not regular, and a metrizable space is always regular. The converse holds only in part: a metrizable space is regular and Hausdorff, and it is second-countable exactly when it is separable.

Going deeper

For a student

The theorem is the first answer to the metrization problem: which topological spaces come from a metric. The proof is an embedding into the Hilbert cube, a standard method that also shows each such space is a subspace of one fixed compact metric space. It characterizes separable metrizable spaces as the second-countable regular Hausdorff spaces. The Nagata–Smirnov metrization theorem and the Bing metrization theorem remove the countability condition and give a full characterization.