proximity generated by uniformity

Definition. Let X be a uniform space with uniformity 𝒰. The uniform proximity, or proximity generated by U, is a binary relationMathworldPlanetmath 𝛿 on the subsets of X, given by the formulaMathworldPlanetmathPlanetmath

A𝛿B    U𝒰:U(A×B).

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 𝛿.

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