equivalent condition for being a fundamental system of entourages

Lemma.

Let $X$ be a set and let $\mathcal{B}$ be a nonempty family of subsets of $X\times X$ . Then $\mathcal{B}$ is a fundamental system of entourages of a uniformity on $X$ if and only if it satisfies the following axioms.

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(B1) If $S$ , $T\in\mathcal{B}$ , then $S\cap T$ contains an element of $\mathcal{B}$ .
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(B2) Each element of $\mathcal{B}$ contains the diagonal $\Delta(X)$ .
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(B3) For any $S\in\mathcal{B}$ , the inverse relation of $S$ contains an element of $\mathcal{B}$ .
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(B4) For any $S\in\mathcal{B}$ , there is an element $T\in\mathcal{B}$ such that the relational composition $T\circ T$ is contained in $S$ .

Proof.

Suppose $\mathcal{B}$ is a fundamental system of entourages for a uniformity $\mathcal{U}$ . Verification of axiom (B2) is immediate, since $\mathcal{B}\subseteq\mathcal{U}$ and each entourage is already required to contain the diagonal of $X$ . We will prove that $\mathcal{B}$ satisfies (B1); the proofs that (B3) and (B4) hold are analogous.

Let $S$ , $T$ be entourages in $\mathcal{B}\subseteq\mathcal{U}$ . Since $\mathcal{U}$ is closed under binary intersections, $S\cap T\in\mathcal{U}$ . By the definition of fundamental system of entourages, since $S\cap T\in\mathcal{U}$ , there exists an entourage $B\in\mathcal{B}$ such that $B\subseteq S\cap T$ . Thus $\mathcal{B}$ satisfies axioms (B1) through (B4).

To prove the converse, define a family of subsets of $X\times X$ by

\mathcal{U}=\{S\subseteq X\times X\colon B\subseteq S\text{\ for some $B\in% \mathcal{B}$}\}.

By construction, each element of $\mathcal{U}$ contains an element of $\mathcal{B}$ , so all that remains is to show that $\mathcal{U}$ is a uniformity. Suppose $T$ is a subset of $X\times X$ that contains an element $S\in\mathcal{U}$ . By the definition of $\mathcal{U}$ , there exists some $B\in\mathcal{B}$ such that $B\subseteq S$ . Since $S\subseteq T$ , it follows that $B\subseteq T$ , so $T$ satisfies the requirement for membership in $\mathcal{U}$ . Thus $\mathcal{U}$ is closed under taking supersets. The remaining axioms for a uniformity follow directly from the appropriate axioms for the fundamental system of entourages by applying the axiom we have just checked. Hence $\mathcal{B}$ is a fundamental system of entourages for a uniformity on $X$ . ∎

Title	equivalent condition for being a fundamental system of entourages
Canonical name	EquivalentConditionForBeingAFundamentalSystemOfEntourages
Date of creation	2013-03-22 16:30:02
Last modified on	2013-03-22 16:30:02
Owner	mps (409)
Last modified by	mps (409)
Numerical id	5
Author	mps (409)
Entry type	Derivation
Classification	msc 54E15