# complete uniform space

Let $X$ be a uniform space with uniformity $\mathcal{U}$. A filter $\mathcal{F}$ on $X$ is said to be a *Cauchy filter* if for each entourage $V$ in $\mathcal{U}$, there is an $F\in \mathcal{F}$ such that $F\times F\subseteq V$.

We say that $X$ is *complete ^{}* if every Cauchy filter is a convergent filter in the topology

^{}${T}_{\mathcal{U}}$ induced (http://planetmath.org/TopologyInducedByUniformStructure) by $\mathcal{U}$. $\mathcal{U}$ in this case is called a

*complete uniformity*.

A *Cauchy sequence* $\{{x}_{i}\}$ in a uniform space $X$ is a sequence in $X$ whose section filter is a Cauchy filter. A Cauchy sequence is said to be convergent if its section filter is convergent. $X$ is said to be *sequentially complete* if every Cauchy sequence converges^{} (every section filter of it converges).

Remark. This is a generalization^{} of the concept of completeness in a metric space, as a metric space is a uniform space. As we see above, in the course of this generalization, two notions of completeness emerge: that of completeness and sequentially completeness. Clearly, completeness always imply sequentially completeness. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.

Title | complete uniform space |

Canonical name | CompleteUniformSpace |

Date of creation | 2013-03-22 16:41:40 |

Last modified on | 2013-03-22 16:41:40 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 54E15 |

Synonym | semicomplete |

Synonym | semi-complete |

Related topic | Complete |

Defines | Cauchy filter |

Defines | Cauchy sequence |

Defines | sequentially complete |

Defines | complete uniformity |