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 .