Bourbaki

Proposition 4

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

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

A path-connected topological space is connected.

Since every interval of R\mathbf{R} is connected, the image of a path is connected (TG, I, p. 82, prop. 4). The empty space is connected. Since a nonempty path-connected space is the union of the images of the paths issuing from one of its points, it is connected (TG, I, p. 81, prop. 2).

Taking into account the identification (III, p. 256, remark 1) of the paths in X with the homotopies between mappings of a topological space P reduced to a point into X, Proposition 1 of III, p. 230 gives:

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.

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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.