PlanetMath: generalization of a uniformity

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

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.

(view preamble | get metadata)

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

This object's parent.

Log in to rate this entry.
(view current ratings)

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 4338 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact