# 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 $f:\mathcal{N}\to X\setminus S$ 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 $\mathbb{N}=\{1,2,\mathrm{\dots}\}$ 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)

$$u:\mathcal{N}\coprod \mathbb{N}\to X$$ |

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 |
---|---|

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 |