separated uniform space
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:
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.
. 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 . ∎
|Title||separated uniform space|
|Date of creation||2013-03-22 16:42:34|
|Last modified on||2013-03-22 16:42:34|
|Last modified by||CWoo (3771)|
|Synonym||Hausdorff uniform space|