The Nagata-Smirnov Metrization Theorem gives a necessary and sufficient condition for a topological space to be metrizable, meaning its topology can be induced by some distance function, showing this holds exactly when the space is regular and Hausdorff and has a basis that can be written as a countable union of locally finite collections of open sets. Named for Jun-iti Nagata and Yuri Smirnov, it is one of the principal general metrization theorems of point-set topology.
Facts
Partially Attested
Proof YearNagata's proof was published in 1950 and Smirnov's independent proof in 1951; 1951 is recorded as Value, the year the biconditional theorem carrying both names was complete StatementA topological space X is metrizable if and only if it is regular and has a countably locally finite (that is, sigma-locally finite) basis. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Nagata-Smirnov metrization theorem, Wikipedia
Proved By
Source Nagata-Smirnov metrization theorem, Wikipedia
Sources
1. Nagata-Smirnov metrization theorem, Wikipedia
Statement section
A topological space X is metrizable if and only if it is regular and has a countably locally finite (that is, sigma-locally finite) basis.
Naming section
The theorem is named after Junichi Nagata and Yurii Mikhailovich Smirnov, whose (independent) proofs were published in 1950 and 1951, respectively.
- In Branch: Topology, Lead sentence
Proved By: Jun-iti Nagata, Lead paragraph
In topology, the Nagata-Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological
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