Urysohn's metrization theorem

Urysohn's metrization theorem states that every regular second-countable topological space is metrizable: its topology is induced by a metric.

Regular space: A topological space in which points and closed sets not containing them can be separated by disjoint open sets. Regularity is one of the two topological hypotheses in the theorem.
Urysohn's lemma: A theorem stating that disjoint closed sets in a normal space can be separated by a continuous function into the unit interval. It provides the continuous functions used to distinguish points from closed sets.
Pavel Urysohn: A Russian mathematician whose work in topology included the lemma and metrization theorem bearing his name. His name is attached to both the separation tool and this metrization result.
Nagata–Smirnov metrization theorem: A metrization theorem characterizing metrizable spaces through regularity and a σ-locally finite base. It replaces second countability with a broader base condition and characterizes all metrizable spaces.
Metrizable space: A topological space whose open sets are exactly those generated by some metric. Metrizability is the conclusion the theorem guarantees.
Countable base: A countable family of open sets such that every open neighborhood of a point contains one of them containing that point. The proof uses a countable base to select a countable family of separating functions.
General topology: The branch of mathematics that studies topological spaces and properties preserved by continuous maps. The theorem is a central bridge from abstract topological axioms to metric structure.
Bing metrization theorem: A metrization theorem characterizing metrizable spaces through regularity and a σ-discrete base. It offers a base-based criterion that extends beyond Urysohn’s countable-base hypothesis.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. The theorem asks whether a topology can arise from this distance-based structure.
Continuous function: A function between topological spaces for which inverse images of open sets are open. The constructed functions encode topological separation as real-valued data.

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