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 |
---|---|
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 |