# Hausdorff property is hereditary

###### Theorem 1.

A subspace^{} of a Hausdorff space is Hausdorff.

###### Proof.

Let $X$ be a Hausdorff space , and let $Y$ be a subspace of $X$. Let ${y}_{1},{y}_{2}\in Y$ where ${y}_{1}\ne {y}_{2}$.
Since $X$ is Hausdorff, there are disjoint neighborhoods^{} ${U}_{1}$ of ${y}_{1}$ and ${U}_{2}$ of ${y}_{2}$ $\in X$. Then
${U}_{1}\cap Y$ is a neighborhood of ${y}_{1}$ in $Y$ and ${U}_{2}\cap Y$ is a neighborhood of ${y}_{2}$ in $Y$, and ${U}_{1}\cap Y$ and ${U}_{2}\cap Y$ are disjoint.
Therefore, $Y$ is Hausdorff.
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Title | Hausdorff property is hereditary |
---|---|

Canonical name | HausdorffPropertyIsHereditary |

Date of creation | 2013-03-22 15:22:27 |

Last modified on | 2013-03-22 15:22:27 |

Owner | georgiosl (7242) |

Last modified by | georgiosl (7242) |

Numerical id | 8 |

Author | georgiosl (7242) |

Entry type | Theorem |

Classification | msc 54D10 |