Remark 1

For every finite open covering $\mathscr{R} = (U_i)_{1 \leq i \leq n}$ of $X$, the set

is a neighbourhood of $\Delta$ in $X \times X$, and these sets $V_{\mathscr{R}}$ form a fundamental system of neighbourhoods of $\Delta$ (and therefore a fundamental system of entourages of the unique uniformity on $X$). Let $W$ be any neighbourhood of $\Delta$ in $X \times X$; then for each $x \in X$ there is an open neighbourhood $U_x$ of $x$ in $X$ such that $U_x \times U_x \subset W$. Since the $U_x$ ($x \in X$) form an open covering of $X$, there exist a finite number of points $x_i$ ($1 \leq i \leq n$) such that the $U_{x_i}$ ($1 \leq i \leq n$) form a covering $\mathscr{R}$ of $X$. We have then $V_{\mathscr{R}} \subset W$, which proves the assertion.

For this reason the unique uniformity on $X$ is often called the uniformity of finite open coverings (cf. Chapter IX, § 4, Exercise 17).