Bourbaki

Proposition 14

Tag 01XO, permanent. Cite it as [Bourbaki, Tag 01XO].

Topologie algébrique, chapter III, § 2. Homotopie et chemins

Every compact, metrizable, nonempty, totally disconnected topological space without isolated point is homeomorphic to {0,1}N\{0,1\}^{\mathbf{N}}.

Let X be such a topological space. Let us endow it with a distance compatible with its topology. Since the space X is compact, it is complete for this distance (TG, II, p. 27, th. 1).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

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Cites

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Machine translation of the English, by gpt-5-6-mini, gpt-5-mini, gpt-5-6, gpt-5.4, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.