# generalization of a uniformity

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

1. 1.

for every $U\in\mathcal{U}$, $U^{-1}\in\mathcal{U}$

2. 2.

for every $U\in\mathcal{U}$, there is $V\in\mathcal{U}$ 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 $\mathcal{U}$ satisfies Axiom 1, then $\mathcal{U}$ is called a semi-uniformity. If $\mathcal{U}$ satisfies Axiom 2, then $\mathcal{U}$ is called a quasi-uniformity. The underlying set $X$ equipped with $\mathcal{U}$ is called a semi-uniform space or a quasi-uniform space according to whether $\mathcal{U}$ 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.

## References

• 1 W. Page, Topological Uniform Structures, Wiley, New York 1978.
 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