Definition 3

Let $V$ be a symmetric entourage of a uniform space $X$. A finite sequence $(x_i)_{0 \leq i \leq n}$ of points of $X$ is said to be a $V$-chain if $x_i$ and $x_{i+1}$ are $V$-close for $0 \leq i < n$. The points $x_0$ and $x_n$ are called the ends of the $V$-chain, and they are said to be joined by the $V$-chain.

Given a symmetric entourage $V$, the relation "there is a $V$-chain joining $x$ and $y$" is an equivalence relation between $x$ and $y$ in $X$. Let $A_{x, v}$ be the equivalence class of $x$ for this relation, i.e. the set of all $y \in X$ which can be joined to $x$ by a $V$-chain. Clearly if $y \in A_{x, v}$ then $V(y) \subset A_{x, v}$; hence $A_{x, v}$ is open in $X$; and the complement of $A_{x, v}$, being a union of equivalence classes, is also open. Hence: