Proposition 15

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

General Topology, chapter II, § 3. Complete spaces

*Let X and X' be two uniform spaces. For each uniformly continuous mapping $f : X \to X'$ there is a unique uniformly continuous mapping $\hat{f} : \hat{X} \to \hat{X}'$ such that the diagram

\begin{array}{ccc} X & \xrightarrow{i'} & X' \\ i \downarrow & & \downarrow i' \\ \hat{X} & \xrightarrow{\hat{f}} & \hat{X}' \end{array}

is commutative (*), where $i : X \to \hat{X}$ and $i' : X' \to \hat{X}'$ are the canonical mappings.

Apply Theorem 3 to the function $i' \circ f : X \to \hat{X}'$ .

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

Cited by

Proposition 5, § 2
Corollary, § 3
Proposition 18, § 3
Proposition 2, § 4
Proposition 4, § 2

Cites

Theorem 3