Polish space is a space homeomorphic to a complete metric space with a countable dense subset.
Examples
- Real direct
- Any open or closed subset of a real line
- separable Banach space .
- Cantor Set
- Direct work of Polish spaces Polish.
Properties
- A closed subset of Polish space is Polish.
- An open subset of Polish space is Polish.
- ( Alexandrov 's theorem) Any G-delta-set of Polish space is Polish;
- The converse is also true, if a subset of the Polish space is Polish, then it is a G-delta set.
- ( Alexandrov 's theorem) Any G-delta-set of Polish space is Polish;
- Between any two uncountable Polish spaces there is a Borel bijection . That is, a bijection that translates Borel sets into Borel sets .
- In particular, every uncountable Polish space has a continuum power .
- ( Cantor-Bendixson theorem ) any closed subset in Polish space is represented as a disjoint union of a perfect subset , a countable and an open subset.
Literature
- V. G. Canovey, V. A. Lyubetskiy. Modern set theory: Borel and projective sets . - ICMMO, 2010 .-- 320 p. - ISBN 78-5-94057-683-9.