Proposition 6

Tag 01X1, permanent. Cite it as [Bourbaki, Tag 01X1].

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

Let $(Y_j)_{j\in J}$ be a family of topological spaces. The mapping

$(\pi_0$ (pr $_j$ )) $:\pi_0(\prod_{j\in J}Y_j)\rightarrow \prod_{j\in J}\pi_0(Y_j)$

is bijective. In particular, the product space of a family of path-connected spaces is path-connected.

This follows from Proposition 3 of III, p. 231 where one takes for X a space reduced to a point.

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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