# T3 space

A *regular space ^{}* is a topological space

^{}in which points and closed sets can be separated by open sets; in other words, given a closed set $A$ and a point $x\notin A$, there are disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subseteq V$.

A *${\mathrm{T}}_{\mathrm{3}}$ space* is a regular^{} ${\mathrm{T}}_{0}$-space (http://planetmath.org/T0Space).
A ${\mathrm{T}}_{3}$ space is necessarily also ${\mathrm{T}}_{2}$, that is, Hausdorff^{}.

Note that some authors make the opposite distinction between ${\mathrm{T}}_{3}$ spaces and regular spaces, that is, they define ${\mathrm{T}}_{3}$ spaces to be topological spaces in which points and closed sets can be separated by open sets, and then define regular spaces to be topological spaces that are both ${\mathrm{T}}_{3}$ and ${\mathrm{T}}_{0}$. (With these definitions, ${\mathrm{T}}_{3}$ does not imply ${\mathrm{T}}_{2}$.)

Title | T3 space |

Canonical name | T3Space |

Date of creation | 2013-03-22 12:18:24 |

Last modified on | 2013-03-22 12:18:24 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 14 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D10 |

Related topic | Tychonoff^{} |

Related topic | T2Space |

Related topic | T1Space |

Related topic | T0Space |

Defines | T3 |

Defines | regular |

Defines | regular space |