Proposition 5

In a uniform space $X$, the set $A_{x, v}$ of points which can be joined by a $V$-chain to a given point $x$ is both open and closed in $X$.

For each $x \in X$, let $A_x$ denote the intersection of the sets $A_{x, v}$ as $V$ runs through the set of symmetric entourages of $X$; $A_x$ is the equivalence class of $x$ for the equivalence relation "for all symmetric entourages $V$, there is a $V$-chain joining $x$ and $y$".