# uncountable Polish spaces contain Cantor space

Cantor space is an example of a compact^{} and uncountable Polish space^{}. In fact, every uncountable Polish space contains Cantor space, as stated by the following theorem.

###### Theorem.

Let $X$ be an uncountable Polish space. Then, it contains a subset $S$ which is homeomorphic to Cantor space.

For example, the set $\mathbb{R}$ of real numbers contains the Cantor middle thirds set (http://planetmath.org/CantorSet). Note that, being homeomorphic to Cantor space, $S$ must be a compact and hence closed subset of $X$.
The result is trivial in the case of Baire space^{} $\mathcal{N}$, in which case we may take $S$ to be the set of all $s\in \mathcal{N}$ satisfying ${s}_{n}\in \{1,2\}$ for all $n$.
Then, for any uncountable Polish space $X$ there exists a continuous^{} and one-to-one function $f:\mathcal{N}\to X$ (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Then $f$ gives a continuous bijection from $S$ to $f(S)$. The inverse function theorem (http://planetmath.org/InverseFunctionTheoremTopologicalSpaces) implies that $f$ is a homeomorphism between $S$ and $f(S)$ and, therefore, $f(S)$ is homeomorphic to Cantor space.

Title | uncountable Polish spaces contain Cantor space |
---|---|

Canonical name | UncountablePolishSpacesContainCantorSpace |

Date of creation | 2013-03-22 18:48:33 |

Last modified on | 2013-03-22 18:48:33 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 54E50 |

Related topic | PolishSpace |