Uniform space in the entourage formulation

Let $X$ be a set and write $\Delta_X=\{(x,x):x\in X\}$ for its diagonal (The diagonal $\Delta_X \subseteq X \times X$, the diagonal map $\delta_X$, and the pairing $\langle f, g \rangle$ of two maps). For $E\subseteq X\times X$, put $E[x]:=\{y\in X:(x,y)\in E\}$, $E^{-1}:=\{(y,x):(x,y)\in E\}$, and $E\circ F:=\{(x,z):\text{some }y\text{ has }(x,y)\in E,(y,z)\in F\}$.

A uniformity on $X$ is a filter $\mathcal U$ on $X\times X$ (Filter on a set) such that:

• every $E\in\mathcal U$ contains $\Delta_X$;
• $E\in\mathcal U$ implies $E^{-1}\in\mathcal U$;
• for every $E\in\mathcal U$ there is $D\in\mathcal U$ with $D\circ D\subseteq E$.

Its members are entourages. A uniform space is a set equipped with a uniformity. The induced topology and its neighbourhoods are constructed in The sets containing an entourage ball about each of their points form a topology.