# Tychonoff space

A topological space^{} $X$ is said to be *completely regular ^{}* if whenever $C\subseteq X$ is closed and $x\in X\setminus C$ then there is a continuous function

^{}$f:X\to [0,1]$ with $f(x)=0$ and $f(C)\subseteq \{1\}$.

A completely regular space that is also ${T}_{0}$ (http://planetmath.org/T0Space) (and therefore Hausdorff^{} (http://planetmath.org/T2Space))
is called a *Tychonoff space*, or a *${T}_{\mathrm{3}\mathrm{\u2064}\frac{\mathrm{1}}{\mathrm{2}}}$ space*.

Some authors interchange the meanings of ‘completely regular’ and ‘${T}_{3\u2064\frac{1}{2}}$’ compared to the above.

It can be proved that a topological space is Tychonoff if and only if it has a Hausdorff compactification.

Title | Tychonoff space |

Canonical name | TychonoffSpace |

Date of creation | 2013-03-22 12:12:42 |

Last modified on | 2013-03-22 12:12:42 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D15 |

Synonym | Tikhonov space |

Synonym | Tychonoff topological space |

Synonym | Tikhonov topological space |

Synonym | Tychonov space |

Synonym | Tychonov topological space |

Related topic | NormalTopologicalSpace |

Related topic | T3Space |

Defines | Tychonoff |

Defines | completely regular |

Defines | completely regular space |

Defines | Tikhonov |

Defines | Tychonov |