uniform neighborhood

Let X be a uniform space with uniformity 𝒰. For each x∈X and Uβˆˆπ’°, define the following items

  • β€’

    U⁒[x]:={y∣(x,y)∈U}, and

  • β€’


  • β€’


PropositionPlanetmathPlanetmath. 𝔑x is the abstract neighborhood system around x, hence 𝔑 is the abstract neighborhood system of X.


We show that all five defining conditions of a neighborhood system on a set are met:

  1. 1.

    For each (x,U⁒[x])βˆˆπ”‘, x∈U⁒[x], since every entourage contains the diagonal relation.

  2. 2.

    Every x∈X and every entourage Uβˆˆπ’°, U⁒[x]βŠ†X with (x,U⁒[x])βˆˆπ”‘

  3. 3.

    Suppose (x,U⁒[x])βˆˆπ”‘ and U⁒[x]βŠ†YβŠ†X. Showing that (x,Y)βˆˆπ”‘ amounts to showing Y=V⁒[x] for some Vβˆˆπ’°. First, note that each entourage U can be decomposed into disjoint unionMathworldPlanetmath of sets β€œslices” of the form {a}Γ—U⁒[a]. We replace the β€œslice” {x}Γ—U⁒[x] by {x}Γ—Y. The resulting disjoint union is a set V, which is a supersetMathworldPlanetmath of U. Since 𝒰 is a filter, Vβˆˆπ’°. Furthermore, V⁒[x]=Y.

  4. 4.

    a∈U⁒[x]∩V⁒[x] iff (x,a)∈U∩V iff a∈(U∩V)⁒[x]. This implies that if (x,U⁒[x]),(x,V⁒[x])βˆˆπ”‘, then (x,U⁒[x]∩V⁒[x])=(x,(U∩V)⁒[x])βˆˆπ”‘.

  5. 5.

    Suppose (x,U⁒[x])βˆˆπ”‘. There is Vβˆˆπ’° such that (V∘V)⁒[x]βŠ†U⁒[x]. We show that V⁒[x]βŠ†X is what we want. Clearly, x∈V⁒[x]. For any y∈V⁒[x], and any a∈V⁒[y], we have (x,a)=(x,y)∘(y,a)∈V∘V, or a∈(V∘V)⁒[x]βŠ†U⁒[x]. So V⁒[y]βŠ†U⁒[x] for any y∈V⁒[x]. In order to show that (y,U⁒[x])βˆˆπ”‘, we must find Wβˆˆπ’° such that U⁒[x]=W⁒[y]. By the third step above, since V⁒[y]βŠ†U⁒[x], there is Wβˆˆπ’° with W⁒[y]=U⁒[x]. Thus (y,U⁒[x])=(y,W⁒[y])βˆˆπ”‘.


Definition. For each x in a uniform space X with uniformity 𝒰, a uniform neighborhood of x is a set U⁒[x] for some entourage Uβˆˆπ’°. In general, for any AβŠ†X, the set

U⁒[A]:={y∈X∣(x,y)∈U⁒ for some ⁒x∈A}

is called a uniform neighborhood of A.

Two immediate properties that we have already seen in the proof above are: (1). for each Uβˆˆπ’°, x∈U⁒[x]; and (2). U⁒[x]∩V⁒[x]=(U∩V)⁒[x]. More generally, β‹‚Ui⁒[x]=(β‹‚Ui)⁒[x].

Remark. If we define T𝒰:={AβŠ†Xβˆ£βˆ€x∈A,βˆƒUβˆˆπ’°β’Β such that ⁒U⁒[x]βŠ†A}, then T𝒰 is a topologyMathworldPlanetmath induced by the uniform structure (http://planetmath.org/TopologyInducedByAUniformStructure) 𝒰. Under this topology, uniform neighborhoods are synonymous with neighborhoodsMathworldPlanetmathPlanetmath.

Title uniform neighborhood
Canonical name UniformNeighborhood
Date of creation 2013-03-22 16:42:31
Last modified on 2013-03-22 16:42:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54E15
Related topic TopologyInducedByAUniformStructure