proof of Polish spaces up to Borel isomorphism
We show that every uncountable Polish space is Borel isomorphic to the real numbers. First, there exists a continuous one-to-one and injective function from Baire space to such that is countable, and such that the inverse from to is Borel measurable (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Letting be any countably infinite subset of , the same result can be applied to , which is also a Polish space. So, there is a continuous and one-to-one function such that is countable and such that the inverse defined on is Borel. Then, contains and is countably infinite. Hence, there is a invertible function from to . Under the discrete topology on this is necessarily a continuous function with Borel measurable inverse. By combining the functions and , this gives a continuous, one-to-one and onto function from the disjoint union (http://planetmath.org/TopologicalSum)
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 from to with Borel inverse. So, gives the desired Borel isomorphism from to .
|Title||proof of Polish spaces up to Borel isomorphism|
|Date of creation||2013-03-22 18:47:18|
|Last modified on||2013-03-22 18:47:18|
|Last modified by||gel (22282)|