# 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 $\mathcal{N}$ to $X$ such that $X\setminus f(\mathcal{N})$ is countable, and such that the inverse from $f(\mathcal{N})$ to $\mathcal{N}$ 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 $f\colon\mathcal{N}\rightarrow X\setminus S$ such that $S^{\prime}\equiv X\setminus f(\mathcal{N})$ 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 $\mathbb{N}=\{1,2,\ldots\}$ 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)

 $u\colon\mathcal{N}\coprod\mathbb{N}\rightarrow X$

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

Title proof of Polish spaces up to Borel isomorphism ProofOfPolishSpacesUpToBorelIsomorphism 2013-03-22 18:47:18 2013-03-22 18:47:18 gel (22282) gel (22282) 6 gel (22282) Proof msc 54E50