NagataSmirnov

Nagata–Smirnov metrization theorem is a fundamental result in general topology that provides a precise criterion for when a topological space can be given a metric. The theorem is named after the mathematicians Nagata and Smirnov.

Statement: If X is a topological space, then X is metrizable (its topology arises from a metric) if and only if X is regular and has a base that is a countable union of locally finite families, also described as a sigma-locally finite base. A base is locally finite if every point has a neighborhood that intersects only finitely many sets from the base; a sigma-locally finite base is a base that can be written as a countable union of locally finite families.

Remarks: Regularity (often denoted T3) is a separation condition ensuring the ability to separate points from closed sets by neighborhoods. The theorem thus links a separation property with a combinatorial condition on the base of open sets, providing a practical metrization criterion. It is one of several classical metrization theorems in topology, alongside Urysohn’s and Bing’s criteria, and sits alongside Moore’s metrization concepts related to developable spaces.

Applications and examples: The theorem applies to many familiar spaces, including all separable metric spaces, and provides a tool for proving metrizability by verifying the sigma-locally finite base property. It also clarifies why certain spaces fail to be metrizable despite possessing several intuitive features.

See also: Urysohn metrization theorem, Bing metrization theorem, Moore’s metrization conjecture, general topology references.