This is established in the proof of the Urysohn metrization theorem.

This would have been a nice metrization theorem.

The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.

For a closely related theorem see the Bing metrization theorem.

Urysohn's metrization theorem states that every second-countable, regular space is metrizable.

The Nagata-Smirnov metrization theorem extends this to the non-separable case.

The study of locally discrete collections is worthwhile as Bing's metrization theorem shows.

Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems.

The following is known as Bing's metrization theorem:

Several other metrization theorems follow as simple corollaries to Urysohn's Theorem.

Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem.

Such spaces are exactly the separable and metrizable X (see Urysohn's metrization theorem).

Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

The Nagata-Smirnov metrization theorem, described below, provides a more specific theorem where the converse does hold.

The Nagata-Smirnov metrization theorem in topology characterizes when a topological space is metrizable.

Therefore, if X is a metrizable space with a countable basis, one implication of Bing's metrization theorem holds.

Without assuming Urysohn's metrization theorem, one can prove that every regular space with a countable base is a G space.

There are numerous characterizations that tell when a second countable topological space is metrizable, such as Urysohn's metrization theorem.

In 1951 he proved results regarding the metrizability of topological spaces, including what would later be called the Bing-Nagata-Smirnov metrization theorem.

In fact, Bing's metrization theorem is almost a corollary of the Nagata-Smirnov theorem.

The first really useful metrization theorem was Urysohn's metrization theorem.

In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theorem, second-countable then implies metrizable.

The Stone's metrization theorem has been named after him, and he was a member of a group of mathematicians who published pseudonymously as Blanche Descartes.

A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.

His name is attached to the Nagata-Smirnov metrization theorem, which was proved independently by Nagata in 1950 and by Smirnov in 1951.