Urysohn's Metrizability Theorem

The Metrisierbarkeitssatz Urysohn - or Metrisationssatz Urysohn ( English Urysohn's theorem metrization ) - is a classic mathematical theorem in the field of topology , which the Russian mathematician Paul Urysohn back. The theorem deals with the question of the metrizability of topological spaces in connection with countability conditions . According to the mathematician Lutz Führer , the metrizability theorem is one of the most famous results of P. Urysohn .

Formulation of the sentence

The sentence can be summarized as follows:

For a Hausdorff space that satisfies the second axiom of countability , regularity , complete regularity , normality and metrizability are equivalent properties.

It even applies:

For a T1 room , the following conditions are equivalent: ${\displaystyle X}$

(1) is a regular space and satisfies the second axiom of countability. ${\displaystyle X}$

(2) is a separable and metrizable space . ${\displaystyle X}$

(3) can be embedded in the Hilbert cube . ${\displaystyle X}$ ${\displaystyle I^{\aleph _{0}}}$

Corollaries

From Urysohn's metrisability theorem there are three direct consequences:

(1) A compact Hausdorff space is metrizable if and only if it satisfies the second axiom of countability.

(2) A locally compact Hausdorff space that satisfies the second axiom of countability is a σ-compact space and as such - just like its one-point compactification - can be metrized.

(3) The constant image of a compact metric space in a Hausdorff space is always a metrizable space.

See also

• Bing Nagata Smirnov theorem

literature

• Lutz Führer: General topology with applications . Vieweg Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 . MR0423277
• Paul Urysohn: On the problem of metrization . In: Mathematical Annals . tape 94 , 1925, pp. 309-315 ( [1] ).
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts (et al.) 1970. MR0264581

Individual evidence

1. ↑ ^{a b} Horst Schubert: Topology. 1975, p. 97
2. ↑ ^{a b c} Stephen Willard: General Topology. 1970, p. 166
3. ↑ ^{a b c d} Guide: General Topology with Applications. 1977, p. 132.
4. ↑ According to Lutz Führer, this result goes back to Paul Alexandroff .