proof of Polish spaces up to Borel isomorphism

We show that every uncountable Polish space $X$ is Borel isomorphic to the real numbers. First, there exists a continuous one-to-one and injective function $f$ from Baire space $𝒩$ to $X$ such that $X\setminus f(𝒩)$ is countable, and such that the inverse from $f(𝒩)$ to $𝒩$ is Borel measurable (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Letting $S$ be any countably infinite subset of $X$, the same result can be applied to $X\setminus S$, which is also a Polish space. So, there is a continuous and one-to-one function $f:𝒩\to X\setminus S$ such that $S^{\prime }\equiv X\setminus f(𝒩)$ is countable and such that the inverse defined on $X\setminus S^{\prime }$ is Borel. Then, $S^{\prime }$ contains $S$ and is countably infinite. Hence, there is a invertible function $g$ from $\mathbb{N}=\{1,2,\mathrm{\dots }\}$ to $S^{\prime }$. Under the discrete topology on $\mathbb{N}$ 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 union (http://planetmath.org/TopologicalSum)

$$u:𝒩\coprod \mathbb{N}\to X$$

with Borel measurable inverse. Similarly, the set of real numbers $ℝ$ with the standard topology is an uncountable Polish space and, therefore, there is a continuous function $v$ from $𝒩\coprod \mathbb{N}$ to $ℝ$ with Borel inverse. So, $v\circ u^{-1}$ gives the desired Borel isomorphism from $X$ to $ℝ$.

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