A property of uniform spaces

Question

Is it true that $E\circ E\subseteq E$ for every entourage $E$ of every uniform space?

Answer 1

As noted in the comments, it clearly is not true in general.

If an entourage $E$ has the property that $E\circ E\subseteq E$, then $E$ is a transitive, reflexive relation on $X$, and the entourage $E\cap E^{-1}$ is an equivalence relation on $X$. A uniform space whose uniformity has a base of equivalence relations is non-Archimedean, the uniform analogue of a non-Archimedean metric space; such a space is zero-dimensional, since it clearly has a base of clopen sets. Conversely, it’s known that every zero-dimensional topology is induced by a non-Archimedean uniformity. (I’ve seen this last result cited to B. Banaschewski, Über nulldimensionale Räume, Math. Nachr. $\bf13$ ($1955$), $129$-$140$, and to A.F. Monna, Remarques sur les métriques non-archimédiennes, Indag. Math. $\bf12$ ($1950$), $122$-$133$; ibid. $\bf12$ ($1950$), $179$-$191$, but I’ve not seen these papers.)