# separable space

## Definition

A topological space^{} is said to be *separable ^{}*
if it has a countable

^{}dense subset.

## Properties

All second-countable spaces are separable. A metric space is separable if and only if it is second-countable.

A continuous^{} image of a separable space is separable.

An open subset of a separable space is separable (in the subspace topology).

A product^{} (http://planetmath.org/ProductTopology) of ${2}^{{\mathrm{\aleph}}_{0}}$ or fewer separable spaces
is separable. This is a special case of the Hewitt-Marczewski-Pondiczery Theorem.

A Hilbert space^{} is separable if and only if it has a countable orthonormal basis^{}.

Title | separable space |

Canonical name | SeparableSpace |

Date of creation | 2013-03-22 12:05:45 |

Last modified on | 2013-03-22 12:05:45 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D65 |

Synonym | separable topological space |

Related topic | SecondCountable |

Related topic | Lindelof |

Related topic | EverySecondCountableSpaceIsSeparable |

Related topic | HewittMarczewskiPondiczeryTheorem |

Defines | separable |