fundamental system of entourages

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

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

$$ℬ=\{U\cap U^{-1}:U\in 𝒰\},$$

where $U^{-1}$ denotes the inverse relation of $U$. Since $𝒰$ is closed under taking relational inverses and binary intersections, $ℬ\subseteq 𝒰$. By construction, each $U\in 𝒰$ contains the element of $U\cap U^{-1}\in ℬ$.

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

• (B1) If $S$, $T\in ℬ$, then $S\cap T$ contains an element of $ℬ$.

• (B2) Each element of $ℬ$ contains the diagonal $\mathrm{\Delta }(X)$.

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

• (B4) For any $S\in ℬ$, there is an element $T\in ℬ$ such that the relational composition $T\circ T$ is contained in $S$.

Suppose $ℬ$ is a fundamental system of entourages for uniformities $𝒰$ and $𝒱$. Then $𝒰\subset 𝒱$. To see this, suppose $S\in 𝒰$. Since $ℬ$ is a fundamental system of entourages for $𝒰$, there is some element $B\in ℬ$ such that $B\subset S$. But $ℬ\subset 𝒱$, so $B\in 𝒱$. Hence by applying the fact that $𝒱$ is closed under taking supersets we may conclude that $S\in 𝒱$. So if $ℬ$ 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 $ℬ$.

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