# injective images of Baire space

Every uncountable Polish space^{} is, up to a countable^{} subset, an injective image of Baire space $\mathcal{N}$.

###### Theorem.

Let $X$ be an uncountable Polish space. Then, there is a one-to-one and continuous function^{} $f\mathrm{:}\mathrm{N}\mathrm{\to}X$ such that $X\mathrm{\setminus}f\mathit{}\mathrm{(}\mathrm{N}\mathrm{)}$ is countable.

Although the inverse^{} ${f}^{-1}:f(\mathcal{N})\to \mathcal{N}$ will not generally be continuous, it is at least Borel measurable. It can be shown that this is true for all one-to-one and continuous functions between Polish spaces, although here it follows directly from the construction of $f$ (http://planetmath.org/ProofOfInjectiveImagesOfBaireSpace).

Title | injective images of Baire space |
---|---|

Canonical name | InjectiveImagesOfBaireSpace |

Date of creation | 2013-03-22 18:47:12 |

Last modified on | 2013-03-22 18:47:12 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 54E50 |

Related topic | BaireSpaceIsUniversalForPolishSpaces |

Related topic | SpacesHomeomorphicToBaireSpace |