generalization of a uniformity
Let $X$ be a set. Let $\mathrm{\pi \x9d\x92\xb0}$ be a family of subsets of $X\Gamma \x97X$ such that $\mathrm{\pi \x9d\x92\xb0}$ is a filter, and that every element of $\mathrm{\pi \x9d\x92\xb0}$ contains the diagonal relation $\mathrm{\Xi \x94}$ (reflexive^{}). Consider the following possible βaxiomsβ:

1.
for every $U\beta \x88\x88\mathrm{\pi \x9d\x92\xb0}$, ${U}^{1}\beta \x88\x88\mathrm{\pi \x9d\x92\xb0}$

2.
for every $U\beta \x88\x88\mathrm{\pi \x9d\x92\xb0}$, there is $V\beta \x88\x88\mathrm{\pi \x9d\x92\xb0}$ such that $V\beta \x88\x98V\beta \x88\x88U$,
where ${U}^{1}$ is defined as the inverse relation (http://planetmath.org/OperationsOnRelations) of $U$, and $\beta \x88\x98$ is the composition of relations (http://planetmath.org/OperationsOnRelations). If $\mathrm{\pi \x9d\x92\xb0}$ satisfies Axiom 1, then $\mathrm{\pi \x9d\x92\xb0}$ is called a semiuniformity. If $\mathrm{\pi \x9d\x92\xb0}$ satisfies Axiom 2, then $\mathrm{\pi \x9d\x92\xb0}$ is called a quasiuniformity. The underlying set $X$ equipped with $\mathrm{\pi \x9d\x92\xb0}$ is called a semiuniform space or a quasiuniform space according to whether $\mathrm{\pi \x9d\x92\xb0}$ is a semiuniformity or a quasiuniformity.
A semipseudometric space is a semiuniform space. A quasipseudometric space is a quasiuniform space.
A uniformity is one that satisfies both axioms, which is equivalent^{} to saying that it is both a semiuniformity and a quasiuniformity.
References
 1 W. Page, Topological Uniform Structures, Wiley, New York 1978.
Title  generalization^{} of a uniformity 
Canonical name  GeneralizationOfAUniformity 
Date of creation  20130322 16:43:09 
Last modified on  20130322 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  semiuniform 
Synonym  quasiuniform 
Synonym  semiuniform 
Synonym  quasiuniform 
Related topic  GeneralizationOfAPseudometric 
Defines  semiuniformity 
Defines  quasiuniformity 
Defines  semiuniform space 
Defines  quasiuniform space 