PlanetMath: generalization of a uniformity

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generalization of a uniformity

(Definition)

Let 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”:

for every $U\in \mathcal{U}$ , $U^{-1}\in \mathcal{U}$
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 of , and $\circ$ is the composition of relations. 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 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.

Bibliography

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

"generalization of a uniformity" is owned by CWoo.

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Other names:

semiuniformity, quasiuniformity, semiuniform space, quasiuniform space, semi-uniform, quasi-uniform, semiuniform, quasiuniform

Also defines:

semi-uniformity, quasi-uniformity, semi-uniform space, quasi-uniform space

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Cross-references: equivalent, uniformity, quasi-pseudometric space, semi-pseudometric space, axiom, Reflexive, diagonal relation, contains, filter, subsets
There is 1 reference to this entry.

This is version 2 of generalization of a uniformity, born on 2007-02-20, modified 2007-04-21.
Object id is 8937, canonical name is GeneralizationOfAUniformity.
Accessed 3862 times total.

Classification:

AMS MSC:

54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations)

Pending Errata and Addenda

None.

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