Nagata–Smirnov metrization theorem

The Nagata–Smirnov metrization theorem states that a topological space is metrizable if and only if it is regular and has a σ-locally finite base.

Metrizable space: A topological space whose topology is induced by a metric. This is exactly the class of spaces characterized by the theorem.
Partition of unity: A family of continuous nonnegative functions whose supports are locally finite and whose sum is one. Locally finite covers allow continuous functions to be assembled without uncontrolled infinite sums.
Urysohn metrization theorem: A theorem stating that every regular second-countable space is metrizable. It replaces a σ-locally finite base with the stronger-looking requirement of a countable base.
Second-countable space: A topological space with a countable base. A countable base is σ-locally finite, so regular second-countable spaces satisfy the theorem’s criterion.
Jun-iti Nagata: A Japanese mathematician known for contributions to general topology, including metrization theory. Nagata independently established the metrization criterion bearing his name.
Regular space: A topological space in which points and closed sets not containing them can be separated by disjoint open sets. Regularity is the theorem’s separation condition.
Urysohn lemma: A theorem giving a continuous function separating disjoint closed sets in a normal space. Separation by continuous functions is a standard route from topological structure toward metric constructions.
Bing metrization theorem: A metrization theorem characterizing metrizable spaces through regularity and a σ-discrete base. Its base condition uses discrete families rather than locally finite ones.
Separable metric space: A metric space containing a countable dense subset. Separable metric spaces have countable bases, making the theorem’s condition especially easy to verify.
Yuriĭ Smirnov: A Soviet mathematician known for work in general topology and metrization theory. Smirnov independently proved the same characterization of metrizable spaces.

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