Metrizable space

In the sub-area of topology of mathematics , a metrizable space is a topological space with additional special properties.

Since the metric spaces are special cases of the topological spaces, it makes sense to ask when a topological space can be metrised , that is, which additional requirements a topological space has to meet in order for there to be a metric that induces the topology. This article gives an overview of necessary and sufficient conditions for metrizability , which are explained in more detail in the articles to which reference is made from here. Sentences that weak sufficient conditions or equivalent conditions to formulate Metrisierbarkeit be in the literature as Metrisationssätze referred.

Necessary conditions

Every topological property that metric spaces always fulfill naturally represents a necessary condition for the metrizability of any topological space. Of particular interest, however, are properties that “bring the space close to metrizability”.

• Separation properties: Every metrizable room is a normal room and a Hausdorff room .
• Characteristic of compactness: Every metrizable room is paracompact .
• Countability: Every metrizable space fulfills the first countability axiom .

Sufficient conditions

• Urysohn's metrization theorem ( after Pawel Samuilowitsch Urysohn ): For a Hausdorff space that satisfies the second axiom of countability , regularity , complete regularity , normality and metrizability are equivalent properties.
• Every compact Hausdorff space that satisfies the second axiom of countability is metrizable.
• The product topology of metric spaces can be metrized if the index set is at most countable. ${\displaystyle \textstyle X=\prod _{i\in I}M_{i}}$ ${\displaystyle I}$

Equivalent condition

Nagata-Smirnow's metrization theorem: A topological space is metrizable if and only if it is a regular Hausdorff space and has a σ-locally finite basis . (See Bing-Nagata-Smirnov theorem for a more detailed discussion .)

Metrizability of topological vector spaces

• A topological vector space is metrizable if and only if it satisfies the first axiom of countability . The metric can be chosen so that it is translation-invariant and the open spheres are balanced around the zero point and, if the space is locally convex , convex.
• If there is a countable separating family of semi-norms , the space is locally convex and metrizable .

Fully metrizable rooms

• A topological space is called completely metrizable (also topologically complete ) if it is homeomorphic to a complete metric space.
• There are metric spaces, the underlying metric of which is not a complete metric, but which can still be completely metrized. These include, for example, the open unit interval or the set of irrational numbers.
• According to Hausdorff's theorem , a subspace of a completely metrizable space is completely metrizable if it is a subset . ${\displaystyle G_{\delta }}$ ${\displaystyle G_{\delta }}$
• In general, Čech's theorem applies : A topological space is completely metrizable if and only if it is metrizable and at the same time is topologically complete . A topological space is topologically complete if and only if it is homeomorphic to a set of a compact Hausdorff space . ${\displaystyle G_{\delta }}$
• A fully metrizable locally convex topological vector space is called a Fréchet space .
• A separable, completely metrizable space is called the Polish space , such spaces and in particular their subsets form the subject of investigation in descriptive set theory .

Examples, construction of a metric

The easiest way to construct the metric is when the topological space is a finite product of metric spaces . You can then simply add the metrics, for example: ${\displaystyle X}$ ${\displaystyle (M_{i},d_{i});\;1\leq i\leq n}$

${\displaystyle d((x_{1},x_{2},\dotsc ,x_{n}),(y_{1},y_{2},\dotsc ,y_{n}))=d_{1}(x_{1},y_{1})+d_{2}(x_{2},y_{2})+\dotsb +d_{n}(x_{n},y_{n})}$

One can proceed in a similar way if the topological space is a countable product of metric spaces . Then one must force the convergence of the “infinite sum” through a positive sequence and, if necessary , replace the metrics d _i with topologically equivalent metrics bounded by a common limit. The definition does both: ${\displaystyle X}$ ${\displaystyle (M_{i},d_{i});\;i\in \mathbb {N} }$

${\displaystyle d((x_{i}),(y_{i}))=\sum \limits _{i=0}^{\infty }2^{-i}{\frac {d_{i}(x_{i},y_{i})}{1+d_{i}(x_{i},y_{i})}}}$

Counterexamples

• The product topology of at least two-point metric spaces cannot be metrized if the index set can be overcounted. ${\displaystyle \textstyle X=\prod _{i\in I}M_{i}}$ ${\displaystyle I}$
• The first non-countable ordinal number , provided with its order topology, is not paracompact and therefore cannot be metrised. ${\displaystyle \Omega _{0}}$

literature

• Boto von Querenburg : Set theoretical topology (= university text ). 2nd, revised and expanded edition. Springer, Berlin a. a. 1979, ISBN 3-540-09799-6 .
• Walter Rudin : Functional Analysis (= International Series in Pure and Applied Mathematics ). 2nd edition. McGraw-Hill, New York City NY et al. a. 1991, ISBN 0-07-054236-8 .
• Horst Schubert : Topology. An introduction . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .

Individual evidence

1. ^ Schubert: Topology. 1975, p. 97.
2. ↑ Eduard Čech : On Bicompact Spaces . In: Annals of Mathematics . Vol. 38, No. 4, 1937, pp. 823-844. doi : 10.2307 / 1968839 .
3. ↑ Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. 1970, p. 180 .