Proposition 3

A uniform space $X$ is Hausdorff if and only if the intersection of all the entourages of its uniform structure is the diagonal $\Delta$ of $X \times X$. Every Hausdorff uniform space is regular.

The latter statement follows immediately from Corollary 3 to Proposition 2. We have seen that the closed entourages form a fundamental system of entourages (Proposition 2, Corollary 2); if their intersection is $\Delta$, then $\Delta$ is closed in $X \times X$ and consequently $X$ is Hausdorff (Chapter I, § 8, no. 1, Proposition 1). Conversely, if $X$ is Hausdorff then for every point $(x, y) \notin \Delta$ there is an entourage $V$ of $X$ such that $y \notin V(x)$, or equivalently $(x, y) \notin V$; hence $\Delta$ is the intersection of all the entourages.

If a uniform space $X$ is Hausdorff, we say that the uniform structure of $X$ is Hausdorff. If $\mathfrak{B}$ is a fundamental system of entourages for this structure; then $X$ is Hausdorff if and only if the intersection of all the sets of $\mathfrak{B}$ is $\Delta$.