Proposition 1

In a uniform space every subset of a precompact set, every finite union of precompact sets and the closure of every precompact set are precompact.

The first two assertions are immediate consequences of Theorem 3. Let $X$ be a uniform space, $A$ a precompact subset of $X$, and let $i : X \to \hat{X}$ be the canonical mapping. $i(\overline{A})$ is contained in the closure of $i(A)$ in $\hat{X}$ (Chapter I, § 2, no. 1, Theorem 1), hence the closure of $i(\overline{A})$ in $\hat{X}$ is contained in a compact set and is therefore compact.