separated uniform space
Let be a uniform space with uniformity . is said to be separated or Hausdorff if it satisfies the following separation axiom:
where is the diagonal relation on and is the intersection of all elements (entourages) in . Since , the separation axiom says that the only elements that belong to every entourage of are precisely the diagonal elements . Equivalently, if , then there is an entourage such that .
The reason for calling separated has to do with the following assertion:
is separated iff is a Hausdorff space under the topology induced by (http://planetmath.org/TopologyInducedByAUniformStructure) .
Recall that , where is some uniform neighborhood of where, under , is also a neighborhood of . To say that is Hausdorff under is the same as saying every pair of distinct points in have disjoint uniform neighborhoods.
Proof.
. Suppose is separated and are distinct. Then for some . Pick with . Set , then is symmetric and . Furthermore, . If , then . Since is symmetric, , so , which is a contradiction.
. Suppose is Hausdorff under and for every for some . If , then there are for some . Since by assumption, . But , contradicting the disjointness of and . Therefore . β
Title | separated uniform space |
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Canonical name | SeparatedUniformSpace |
Date of creation | 2013-03-22 16:42:34 |
Last modified on | 2013-03-22 16:42:34 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 5 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54E15 |
Synonym | separating |
Synonym | Hausdorff uniform space |