Corollary

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

General Topology, chapter II, § 3. Complete spaces

Let $X, Y$ be two uniform spaces and $X', Y'$ the associated Hausdorff spaces. For each uniformly continuous mapping $f : X \to Y$ there is a unique uniformly continuous mapping $f' : X' \to Y'$ for which the diagram

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

is commutative, where $i$ and $i'$ are the canonical mappings.

Apply Proposition 16 to $i' \circ f : X \to Y'$ .

The Hausdorff space associated with a uniform space may also be characterized by the following property:

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

Cited by

Remark, § 3

Cites

Proposition 16