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uniform space
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(Definition)
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A uniform structure (or uniformity) on a set  is a non empty set
 of subsets of
 which satisfies the following axioms:
- Every subset of
 which contains a set of
 belongs to
 .
- Every finite intersection of sets of
 belongs to
 .
- Every set of
 is a reflexive relation on  (i.e. contains the diagonal).
- If
 belongs to
 , then
 belongs to
 .
- 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.
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"uniform space" is owned by mps. [ full author list (3) | owner history (1) ]
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(view preamble)
| Also defines: |
uniform structure, uniformity, entourage,  -close, vicinity |
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Cross-references: topological vector spaces, topological groups, metric spaces, Cauchy sequences, uniformly continuous functions, structure, topology induced by uniform structure, topological space, diagonal, reflexive relation, intersection of sets, finite, contains, axioms, subsets, empty set
There are 30 references to this entry.
This is version 9 of uniform space, born on 2002-06-10, modified 2008-06-02.
Object id is 3085, canonical name is UniformSpace.
Accessed 9986 times total.
Classification:
| AMS MSC: | 54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations) |
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Pending Errata and Addenda
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