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

To see that each uniform space $(X,\UU)$ has a fundamental system of entourages, define$$ \BB = \{ U\cap U^{-1} \colon U \in \UU \},$$ where $U^{-1}$ denotes the inverse relation of $U$ . Since $\UU$ is closed under taking relational inverses and binary intersections, $\BB\subseteq\UU$ . By construction, each $U\in\UU$ contains the element of $U\cap U^{-1}\in\BB$ .

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

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

Suppose $\BB$ is a fundamental system of entourages for uniformities $\UU$ and $\VV$ . Then $\UU\subset\VV$ . To see this, suppose $S\in\UU$ . Since $\BB$ is a fundamental system of entourages for $\UU$ , there is some element $B\in\BB$ such that $B\subset S$ . But $\BB\subset\VV$ , so $B\in\VV$ . Hence by applying the fact that $\VV$ is closed under taking supersets we may conclude that $S\in\VV$ . So if $\BB$ is a fundamental system of entourages, it is a fundamental system for a unique uniformity $\UU$ . Thus it makes sense to call $\UU$ the uniformity generated by the fundamental system $\BB$ .

Bibliography

1

Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.