# fundamental system of entourages

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

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

 $\mathcal{B}=\{U\cap U^{-1}\colon U\in\mathcal{U}\},$

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

There is a useful equivalent condition for being a fundamental system of entourages. 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 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 $S$.

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

## References

• 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
Title fundamental system of entourages FundamentalSystemOfEntourages 2013-03-22 16:29:55 2013-03-22 16:29:55 mps (409) mps (409) 5 mps (409) Definition msc 54E15 uniformity generated by