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.
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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 Nagata-Smirnov Metrization Theorem
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Source Nagata-Smirnov metrization theorem, Wikipedia
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1. Nagata-Smirnov metrization theorem, Wikipedia
Lead section
The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich Smirnov, whose (independent) proofs were published in 1950 and 1951, respectively.
Principal Result Of: Jun-iti Nagata, Lead sentence of the article on the result
The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich 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
In Group: General Topology, Lead paragraph
In topology, the Nagata-Smirnov metrization theorem characterizes when a topological space is metrizable.
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