Example

If $\varpi$ is any finite partition $(A_i)_{1 \leq i \leq n}$ of a non-empty set $X$, the set $V_\varpi = \bigcup_i (A_i \times A_i)$ by itself constitutes a fundamental system of entourages of a uniformity $\mathfrak{U}_\varpi$ on $X$ (§ 1, no. 1, Example 2); the uniformity of finite partitions on $X$ (no. 2, Remark 1) is then the least upper bound of the uniformities $\mathfrak{U}_\varpi$.