Corollary 2

The interiors (resp. the closures) of the entourages of $X$ in $X \times X$ form a fundamental system of entourages of $X$.

If $V$ is any entourage of $X$, there is a symmetric entourage $W$ such that $W \subset V$; since $W$ is a neighbourhood of $W$ (Proposition 2), the interior of $V$ in $X \times X$ contains $W$ and is therefore an entourage. Furthermore, we have $W \subset W \subset W \subset V$ by Proposition 2, and hence $V$ contains the closure of an entourage.