generalization of a uniformity

Let X be a set. Let 𝒰 be a family of subsets of XΓ—X such that 𝒰 is a filter, and that every element of 𝒰 contains the diagonal relation Ξ” (reflexiveMathworldPlanetmathPlanetmath). Consider the following possible β€œaxioms”:

  1. 1.

    for every Uβˆˆπ’°, U-1βˆˆπ’°

  2. 2.

    for every Uβˆˆπ’°, there is Vβˆˆπ’° such that V∘V∈U,

where U-1 is defined as the inverse relation ( of U, and ∘ is the composition of relations ( If 𝒰 satisfies Axiom 1, then 𝒰 is called a semi-uniformity. If 𝒰 satisfies Axiom 2, then 𝒰 is called a quasi-uniformity. The underlying set X equipped with 𝒰 is called a semi-uniform space or a quasi-uniform space according to whether 𝒰 is a semi-uniformity or a quasi-uniformity.

A semi-pseudometric space is a semi-uniform space. A quasi-pseudometric space is a quasi-uniform space.

A uniformity is one that satisfies both axioms, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying that it is both a semi-uniformity and a quasi-uniformity.


  • 1 W. Page, Topological Uniform Structures, Wiley, New York 1978.
Title generalizationPlanetmathPlanetmath of a uniformity
Canonical name GeneralizationOfAUniformity
Date of creation 2013-03-22 16:43:09
Last modified on 2013-03-22 16:43:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 54E15
Synonym semiuniformity
Synonym quasiuniformity
Synonym semiuniform space
Synonym quasiuniform space
Synonym semi-uniform
Synonym quasi-uniform
Synonym semiuniform
Synonym quasiuniform
Related topic GeneralizationOfAPseudometric
Defines semi-uniformity
Defines quasi-uniformity
Defines semi-uniform space
Defines quasi-uniform space