Corollary 4

If $\mathfrak{F}$ is a minimal Cauchy filter, then every set of $\mathfrak{F}$ has a non-empty interior which also belongs to $\mathfrak{F}$ (in other words, $\mathfrak{F}$ has a base consisting of open sets).

Let $V$ be any entourage of $X$; then there is an open entourage $U \subset V$ ($\S$ 1, no. 2, Corollary 2 to Proposition 2). For each subset $M$ of $X$, $U(M)$ is open and contained in $V(M)$; hence the result, in view of Proposition 5.