proof of Polish spaces up to Borel isomorphism

We show that every uncountable Polish spaceMathworldPlanetmath X is Borel isomorphic to the real numbers. First, there exists a continuousMathworldPlanetmathPlanetmath one-to-one and injective function f from Baire spacePlanetmathPlanetmath 𝒩 to X such that Xf(𝒩) is countableMathworldPlanetmath, and such that the inversePlanetmathPlanetmathPlanetmath from f(𝒩) to 𝒩 is Borel measurable (see here ( Letting S be any countably infiniteMathworldPlanetmath subset of X, the same result can be applied to XS, which is also a Polish space. So, there is a continuous and one-to-one function f:𝒩XS such that SXf(𝒩) is countable and such that the inverse defined on XS is Borel. Then, S contains S and is countably infinite. Hence, there is a invertible function g from ={1,2,} to S. Under the discrete topology on this is necessarily a continuous function with Borel measurable inverse. By combining the functions f and g, this gives a continuous, one-to-one and onto function from the disjoint unionMathworldPlanetmath (


with Borel measurable inverse. Similarly, the set of real numbers with the standard topology is an uncountable Polish space and, therefore, there is a continuous function v from 𝒩 to with Borel inverse. So, vu-1 gives the desired Borel isomorphism from X to .

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