The Nagata–Smirnov metrization theorem in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
characterizes when a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
. The theorem states that a topological space
is metrizable if and only if it is
regular,
Hausdorff and has a
countably locally finite (that is, -locally finite)
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
.
A topological space
is called a regular space if every non-empty closed subset
of
and a point p not contained in
admit non-overlapping open neighborhoods.
A collection in a space
is countably locally finite (or -locally finite) if it is the union of a countable family of locally finite collections of subsets of
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,
[Y. Smirnov, "A necessary and sufficient condition for metrizability of a topological space" (Russian), ''Dokl. Akad. Nauk SSSR'' 77 (1951), 197–200.] respectively.
See also
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*
*
Notes
References
* .
* .
General topology
Theorems in topology
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