separated uniform space

Let X be a uniform space with uniformity 𝒰. X is said to be separated or HausdorffPlanetmathPlanetmath if it satisfies the following separation axiomMathworldPlanetmath:


where Ξ” is the diagonal relation on X and ⋂𝒰 is the intersectionMathworldPlanetmath of all elements (entourages) in 𝒰. Since Ξ”βŠ†β‹‚π’°, the separation axiom says that the only elements that belong to every entourage of 𝒰 are precisely the diagonal elements (x,x). Equivalently, if xβ‰ y, then there is an entourage U such that (x,y)βˆ‰U.

The reason for calling X separated has to do with the following assertion:

X is separated iff X is a Hausdorff space under the topologyMathworldPlanetmath T𝒰 induced by ( 𝒰.

Recall that T𝒰={AβŠ†X∣for each ⁒x∈A⁒, there is ⁒Uβˆˆπ’°β’, such that ⁒U⁒[x]βŠ†A}, where U⁒[x] is some uniform neighborhood of x where, under T𝒰, U⁒[x] is also a neighborhoodMathworldPlanetmathPlanetmath of x. To say that X is Hausdorff under T𝒰 is the same as saying every pair of distinct points in X have disjoint uniform neighborhoods.


(β‡’). Suppose X is separated and x,y∈X are distinct. Then (x,y)βˆ‰U for some Uβˆˆπ’°. Pick Vβˆˆπ’° with V∘VβŠ†U. Set W=V∩V-1, then W is symmetricPlanetmathPlanetmath and WβŠ†V. Furthermore, W∘WβŠ†V∘VβŠ†U. If z∈W⁒[x]∩W⁒[y], then (x,z),(y,z)∈W. Since W is symmetric, (z,y)∈W, so (x,y)=(x,z)∘(z,y)∈W∘WβŠ†U, which is a contradictionMathworldPlanetmathPlanetmath.

(⇐). Suppose X is Hausdorff under T𝒰 and (x,y)∈U for every Uβˆˆπ’° for some x,y∈X. If xβ‰ y, then there are V⁒[x]∩W⁒[y]=βˆ… for some V,Wβˆˆπ’°. Since (x,y)∈V by assumptionPlanetmathPlanetmath, y∈V⁒[x]. But y∈W⁒[y], contradicting the disjointness of V⁒[x] and W⁒[y]. Therefore x=y. ∎

Title separated uniform space
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