# Tychonoff’s theorem

Let ${({X}_{i})}_{i\in I}$ be a family of nonempty topological spaces^{}. The product space (see product topology)

$$\prod _{i\in I}{X}_{i}$$ |

is compact^{} if and only if each of the spaces ${X}_{i}$ is compact.

Not surprisingly, if $I$ is infinite^{}, the proof requires the Axiom of Choice^{}. Conversely, one can show that Tychonoff^{}’s theorem implies that any product^{} of nonempty sets is nonempty, which is one form of the Axiom of Choice.

Title | Tychonoff’s theorem |
---|---|

Canonical name | TychonoffsTheorem |

Date of creation | 2013-03-22 12:05:14 |

Last modified on | 2013-03-22 12:05:14 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 12 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 54D30 |

Synonym | Tichonov’s theorem |

Related topic | Compact |