Nagata–Smirnov metrization theorem

Short description: Characterizes when a topological space is metrizable

In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space [math]\displaystyle{ X }[/math] is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, 𝜎-locally finite) basis.

A topological space [math]\displaystyle{ X }[/math] is called a regular space if every non-empty closed subset [math]\displaystyle{ C }[/math] of [math]\displaystyle{ X }[/math] and a point p not contained in [math]\displaystyle{ C }[/math] admit non-overlapping open neighborhoods. A collection in a space [math]\displaystyle{ X }[/math] is countably locally finite (or 𝜎-locally finite) if it is the union of a countable family of locally finite collections of subsets of [math]\displaystyle{ X. }[/math]

Unlike Urysohn's metrization theorem, which provides only a sufficient condition for metrizability, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable. The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich Smirnov, whose (independent) proofs were published in 1950 and 1951, respectively.

See also

• Bing metrization theorem – Characterizes when a topological space is metrizable
• Kolmogorov's normability criterion – Characterization of normable spaces
• Uniformizable space – Topological space whose topology is generated by a uniform structure

References

• Munkres, James R. (1975), "Sections 6-2 and 6-3", Topology, Prentice Hall, pp. 247–253, ISBN 0-13-925495-1, https://archive.org/details/topologyfirstcou00munk_0/page/247 .
• Patty, C. Wayne (2009), "7.3 The Nagata–Smirnov Metrization Theorem", Foundations of Topology (2nd ed.), Jones & Bartlett, pp. 257–262, ISBN 978-0-7637-4234-8 .