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 $X$ .∎