Theorem 3

A uniform space $X$ is precompact if and only if, for each entourage $V$ of $X$, there is a finite covering of $X$ by $V$-small sets.

We may express this condition more intuitively by saying that $X$ can be covered by a finite number of sets which are as small as we please.

Let $i : X \to \hat{X}$ be the canonical mapping, then the entourages of $X$ are the inverse images under $i \times i$ of the entourages of $\hat{X}$ (§ 3, no. 7, Proposition 12). Suppose $X$ is precompact, and let $U$ be any entourage of $\hat{X}$; then there is a symmetric entourage $U'$ of $\hat{X}$ such that $\hat{U}' \subset U$. Since $\hat{X}$ is compact, there exist a finite number of points $x_j \in \hat{X}$ such that the $U'(x_j)$ (which are $U$-small) cover $\hat{X}$. If $V$ is the inverse image of $U$ by $i \times i$, the sets $i^{-1}(U'(x_j))$ are $V$-small and cover $X$. Conversely, suppose that for each entourage $V$ of $X$ there is a finite covering of $X$ by $V$-small sets. We have to show that every ultrafilter $\mathfrak{F}$ on $\hat{X}$ is convergent; since $\hat{X}$ is complete, it is enough to show that $\mathfrak{F}$ is a Cauchy filter, i.e. that for each closed entourage $U$ of $\hat{X}$ there is a $U$-small set in $\mathfrak{F}$ (§ 1, no. 2, Proposition 2, Corollary 2). Let $V$ be the inverse image of $U$ under $i \times i$, and let $(B_j)$ be a finite covering of $X$ by $V$-small sets; then the sets $C_j = i(B_j)$ are $U$-small and cover $i(X)$, so that

On the other hand, since $C_j \times C_j \subset U$, and $U$ is closed in $\hat{X} \times \hat{X}$, we have $\overline{C}_j \times \overline{C}_j \subset U$, so that the $\overline{C}_j$ are also $U$-small. Since $\mathfrak{F}$ is an ultrafilter, one of the $\overline{C}_j$ belongs to $\mathfrak{F}$ (Chapter I, § 6, no. 4, Corollary to Proposition 5).

Q.E.D.