Polish spaces up to Borel isomorphism

Two topological spacesMathworldPlanetmath X and Y are Borel isomorphic (http://planetmath.org/BorelIsomorphism) if there is a Borel measurable function f:XY with Borel inversePlanetmathPlanetmathPlanetmath. Such a function is said to be a Borel isomorphism. The following result classifies all Polish spaces up to Borel isomorphism.


Every uncountable Polish spaceMathworldPlanetmath is Borel isomorphic to R with the standard topology.

As the Borel σ-algebra on any countableMathworldPlanetmath metric space is just its power setMathworldPlanetmath, this shows that every Polish space is Borel isomorphic to one and only one of the following.

  1. 1.

    {1,2,,n} for some n0, with the discrete topology.

  2. 2.

    ={1,2,} with the discrete topology.

  3. 3.

    with the standard topology.

In particular, two Polish spaces are Borel isomorphic if and only if they have the same cardinality, and any uncountable Polish space has the cardinality of the continuumMathworldPlanetmathPlanetmath.

Title Polish spaces up to Borel isomorphism
Canonical name PolishSpacesUpToBorelIsomorphism
Date of creation 2013-03-22 18:47:00
Last modified on 2013-03-22 18:47:00
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 54E50
Related topic CategoryOfPolishGroups