# proximity generated by uniformity

Definition. Let $X$ be a uniform space with uniformity $\mathcal{U}$. The *uniform proximity*, or *proximity generated by $\mathrm{U}$*, is a binary relation^{} $\mathit{\delta}$ on the subsets of $X$, given by the formula^{}

$$A\mathit{\delta}B\mathit{\hspace{1em}\hspace{1em}}\iff \mathit{\hspace{1em}\hspace{1em}}\forall U\in \mathcal{U}:U\cap (A\times B)\ne \mathrm{\varnothing}.$$ |

Correctness of the above proximity is given by the below theorem:

Theorem. Proximity generated by a uniformity is a proximity. In other words, $X$ is a proximity space with proximity $\mathit{\delta}$.

Title | proximity generated by uniformity |
---|---|

Canonical name | ProximityGeneratedByUniformity |

Date of creation | 2013-03-22 16:56:20 |

Last modified on | 2013-03-22 16:56:20 |

Owner | porton (9363) |

Last modified by | porton (9363) |

Numerical id | 9 |

Author | porton (9363) |

Entry type | Definition |

Classification | msc 54E17 |

Classification | msc 54E15 |

Classification | msc 54E05 |

Defines | uniform proximity |