Bing Nagata Smirnov theorem

The Bing-Nagata-Smirnow theorem (after RH Bing , J. Nagata and JM Smirnow ) is a theorem from the mathematical sub-area of topology , which characterizes those topological spaces whose topology can be defined by a metric .

The problem

With a first abstraction of the convergence examined in or under investigation , one finds that it is sufficient to have a concept of distance. That casually leads to the concept of metric space. In a further abstraction one only refers to open sets and thus comes to topological space. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

Not every topological space can be metrised : Not every topological space has a metric, so that the open sets of that topology result exactly from the open spheres defined by the distance concept of the metric. It is therefore natural to ask which topological spaces are metrizable, looking for conditions that do not argue about structures or properties that cannot be defined on the basis of arbitrary topological spaces (such as metrics: even in the case of metrizable spaces, this is not possible define the metric of the room). This is the so-called metrization problem , which was open for a long time and was solved by the Bing-Nagata-Smirnov theorem.

Topological terms

The topological terms required to characterize the metric spaces are briefly summarized here. Room classes with purely topological definitions are:

• Hausdorff space : A topological space is Hausdorff space if there are disjoint, open sets with and for two different points . ${\displaystyle X}$ ${\displaystyle x,y\in X}$ ${\displaystyle U,V\subset X}$ ${\displaystyle x\in U}$ ${\displaystyle y\in V}$
• Regular space : A topological space is called regular if there are disjoint, open sets with and for each closed set and each . ${\displaystyle X}$ ${\displaystyle A\subset X}$ ${\displaystyle x\in X\setminus A}$ ${\displaystyle U,V\subset X}$ ${\displaystyle A\subset U}$ ${\displaystyle x\in V}$

The following terms are also of a purely topological nature, i.e. their definitions only use open sets:

• A family of subsets of a topological space is called discrete if for every point there is an open set with and for all but at most one exception. ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle U\subset X}$ ${\displaystyle x\in U}$ ${\displaystyle U\cap V=\emptyset }$ ${\displaystyle V\in {\mathcal {V}}}$

• A family of subsets of a topological space is called locally finite if for every point there is an open set with and for all except at most finitely many exceptions. ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle U\subset X}$ ${\displaystyle x\in U}$ ${\displaystyle U\cap V=\emptyset }$ ${\displaystyle V\in {\mathcal {V}}}$

• A family of subsets of a topological space is called - discrete , if there are countably many discrete systems with . Correspondingly means - locally finite , if there are countably many locally finite systems with . ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {V}}_{n}}$ ${\displaystyle {\mathcal {V}}=\bigcup _{n\in \mathbb {N} }{\mathcal {V}}_{n}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {V}}_{n}}$ ${\displaystyle {\mathcal {V}}=\bigcup _{n\in \mathbb {N} }{\mathcal {V}}_{n}}$

• A family of subsets of a topological space is called a basis of space if each is open and every open set can be written in as a union of sets . ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle V\in {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {V}}}$

Formulation of the sentence

The following theorem from Bing-Nagata-Smirnow solves the metrization problem:

The following statements are equivalent for a topological space : ${\displaystyle X}$

• ${\displaystyle X}$ is metrizable.
• ${\displaystyle X}$is a regular Hausdorff space with a -discrete base. ${\displaystyle \sigma }$
• ${\displaystyle X}$is a regular Hausdorff area with a -local basis. ${\displaystyle \sigma }$

Remarks

Historical remark

The metrisability theorem was found independently from Bing, Nagata and Smirnow in the early 1950s, the version with the -discrete base comes from Bing, the version with the -local base comes from Nagata and Smirnow, also independently. ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Urysohn had already proven special cases in the 1920s :

• A normal space with a countable base is homeomorphic to a subset of the Hilbert space and therefore metrizable. ${\displaystyle \ell ^{2}}$
• A compact Hausdorff space can be metrized if and only if it has a countable basis.

Spaces with a countable basis

An important conclusion from the above sentence by Bing-Nagata-Smirnow is:

For topological spaces with a countable basis , the following statements are equivalent:

1. ${\displaystyle X}$ is metrizable
2. ${\displaystyle X}$is paracompact Hausdorff space
3. ${\displaystyle X}$is normal Hausdorff space
4. ${\displaystyle X}$ is a regular Hausdorff room

The implications 1 2 3 4 are comparatively simple. Since a countable basis is of course -discrete, 4 1 follows from the Bing-Nagata-Smirnow theorem. ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \sigma }$ ${\displaystyle \Rightarrow }$

This theorem is also known as Urysohn's Metrizability Theorem .

Generalizations of metric spaces

The Bing-Nagata-Smirnow theorem led to generalizations of the metric space by weakening the conditions on the properties of the base. A family of subsets of a topological space is called completion-receiving if there is a relationship for each subfamily , and the family is called - completion-receiving if it is a countable union of completion-receiving families. ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}\subset {\mathcal {V}}}$ ${\displaystyle \textstyle \bigcup _{U\in {\mathcal {U}}}{\overline {U}}={\overline {\bigcup _{U\in {\mathcal {U}}}U}}}$ ${\displaystyle \sigma }$

A regular Hausdorff room , which has a -base-preserving-basis, is called a -room . Since -local families -are -closure -preserving, the above theorem of Bing-Nagata-Smirnow shows that -spaces are generalizations of metric spaces. Further weakening of this type leads to further room classes. ${\displaystyle \sigma }$ ${\displaystyle M_{1}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle M_{1}}$

literature

• RH Bing : Metrization of topological spaces. In: Canadian Journal of Mathematics. 3, 1951, ISSN  0008-414X , pp. 175-186, online (PDF; 1.34 MB) .
• Jack G. Ceder: Some generalizations of metric spaces. In: Pacific Journal of Mathematics. 11, 1, 1961, ISSN  0030-8730 , pp. 105-125, online (PDF; 2.01 MB) .
• Wolfgang Franz : Topology. Volume 1: General Topology. de Gruyter, Berlin 1960 ( Göschen Collection 1181, ZDB -ID 842269-2 ).
• Jun-iti Nagata : On a necessary and sufficient condition of metrizability. In: Osaka City University. Journal of the Institute of Polytechnics. Ser. A: Mathematics. 1, 1950, ISSN  0388-0516 , pp. 93-100.
• YM Smirnov : A necessary and sufficient condition for metrizability of a topological space. In: Doklady Akademii Nauk. SSSR. Serija Matematika, Fizika 77, 1951, ZDB -ID 758308-4 , pp. 197-200.