generalization of a uniformity

Let $X$ be a set. Let $𝒰$ be a family of subsets of $X\times X$ such that $𝒰$ is a filter, and that every element of $𝒰$ contains the diagonal relation $\mathrm{\Delta }$ (reflexive). Consider the following possible “axioms”:

1. for every $U\in 𝒰$, $U^{-1}\in 𝒰$

2. for every $U\in 𝒰$, there is $V\in 𝒰$ such that $V\circ V\in U$,

where $U^{-1}$ is defined as the inverse relation (http://planetmath.org/OperationsOnRelations) of $U$, and $\circ$ is the composition of relations (http://planetmath.org/OperationsOnRelations). 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 equivalent to saying that it is both a semi-uniformity and a quasi-uniformity.

Title	generalization 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