Every second-countable space that is also a T3 space is a metrizable space. First regularity of a topological space and the Lindelöf property yield a normal space: cover each of two disjoint closed sets by countably many open sets whose closures avoid the other, and subtract the first opposite closures from the th neighborhood. The resulting unions are disjoint open neighborhoods. Urysohn's lemma then gives, for each pair of basic sets , a continuous function equal to one on the smaller closure and zero outside the larger set. There are countably many such pairs. The shrinking form of regularity of a topological space shows these functions generate the topology, and makes them separate points. Their coordinate map embeds the space into the countable cube, with compatible metric . The bounded tails prove compatibility with the product topology. Thus a countable base converts separation axioms into a numerical distance.
A metrizable space is separable exactly when it is Lindelöf. A countable dense subset gives a countable topological basis of balls with rational radii, hence the open-cover property. Conversely, choose countable subcovers of all radius- ball covers; their centers form a countable dense subset.
