# 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.

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

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

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

• (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 EquivalentConditionForBeingAFundamentalSystemOfEntourages 2013-03-22 16:30:02 2013-03-22 16:30:02 mps (409) mps (409) 5 mps (409) Derivation msc 54E15