Proposition 2

Tag 0BGC, permanent. Cite it as [Bourbaki, Tag 0BGC].

General Topology, chapter II, § 4. Relations between uniform spaces and compact spaces

Let $f : X \to Y$ be a uniformly continuous mapping. If $A$ is any precompact subset of $X$ , then $f(A)$ is a precompact subset of $Y$ .

For if $i : X \to \hat{X}$ and $j : Y \to \hat{Y}$ are the canonical mappings, we have $j(f(A)) = \hat{f}(i(A))$ (§ 3, no. 7, Proposition 15) and hence $j(f(A))$ is relatively compact in $\hat{Y}$ (Chapter I, § 9, no. 4, Theorem 2, Corollary 1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Definition 1, § 4
Proposition 3, § 4
Theorem 1, § 4
Theorem 2 (Ascoli), § 2

Cites

Chapter I, § 9, no. 4, Theorem 2
Corollary 1
§ 3, no. 7, Proposition 15