fundamental system of entourages

Let (X,𝒰) be a uniform space. A subset 𝒰 is a fundamental system of entourages for 𝒰 provided that each entourage in 𝒰 contains an element of .

To see that each uniform space (X,𝒰) has a fundamental system of entourages, define


where U-1 denotes the inverse relation of U. Since 𝒰 is closed underPlanetmathPlanetmath taking relational inversesPlanetmathPlanetmathPlanetmathPlanetmath and binary intersectionsMathworldPlanetmath, 𝒰. By construction, each U𝒰 contains the element of UU-1.

There is a useful equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath condition for being a fundamental system of entourages. Let be a nonempty family of subsets of X×X. Then is a fundamental system of entourages of a uniformity on X if and only if it the following axioms.

  • (B1) If S, T, then ST contains an element of .

  • (B2) Each element of contains the diagonal Δ(X).

  • (B3) For any S, the inverse relation of S contains an element of .

  • (B4) For any S, there is an element T such that the relational compositionPlanetmathPlanetmath TT is contained in S.

Suppose is a fundamental system of entourages for uniformities 𝒰 and 𝒱. Then 𝒰𝒱. To see this, suppose S𝒰. Since is a fundamental system of entourages for 𝒰, there is some element B such that BS. But 𝒱, so B𝒱. Hence by applying the fact that 𝒱 is closed under taking supersets we may conclude that S𝒱. So if is a fundamental system of entourages, it is a fundamental system for a unique uniformity 𝒰. Thus it makes sense to call 𝒰 the uniformity generated by the fundamental system .


  • 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
Title fundamental system of entourages
Canonical name FundamentalSystemOfEntourages
Date of creation 2013-03-22 16:29:55
Last modified on 2013-03-22 16:29:55
Owner mps (409)
Last modified by mps (409)
Numerical id 5
Author mps (409)
Entry type Definition
Classification msc 54E15
Defines uniformity generated by