uniform space
A uniform structure (or uniformity) on a set is a non empty set of subsets of which satisfies the following axioms:
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1.
Every subset of which contains a set of belongs to .
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2.
Every finite intersection of sets of belongs to .
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3.
Every set of is a reflexive relation on (i.e. contains the diagonal).
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4.
If belongs to , then belongs to .
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5.
If belongs to , then exists in such that, whenever , then (i.e. ).
The sets of are called entourages or vicinities. The set together with the uniform structure is called a uniform space.
If is an entourage, then for any we say that and are -close.
Every uniform space can be considered a topological space with a natural topology induced by uniform structure. The uniformity, however, provides in general a richer structure, which formalize the concept of relative closeness: in a uniform space we can say that is close to as is to , which makes no sense in a topological space. It follows that uniform spaces are the most natural environment for uniformly continuous functions and Cauchy sequences, in which these concepts are naturally involved.
Examples of uniform spaces are metric spaces, topological groups, and topological vector spaces.
Title | uniform space |
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Canonical name | UniformSpace |
Date of creation | 2013-03-22 12:46:26 |
Last modified on | 2013-03-22 12:46:26 |
Owner | mps (409) |
Last modified by | mps (409) |
Numerical id | 12 |
Author | mps (409) |
Entry type | Definition |
Classification | msc 54E15 |
Defines | uniform structure |
Defines | uniformity |
Defines | entourage |
Defines | -close |
Defines | vicinity |