Bourbaki

Exercise 13, § 2

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Algebraic Topology, III, p. 333

Let X be a separated topological space; suppose that every point of X possesses a neighbourhood U such that every pair of points of U is joined by an injective path in U. If X is connected, prove (without making use of proposition 18 of III, p. 282) that X is path-connected and that every pair of points of X is joined by an injective path.

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