Proposition 1

Tag 0B29, permanent. Cite it as [Bourbaki, Tag 0B29].

General Topology, chapter I, § 4. Product of topological spaces

Let $f = (f_i)$ be a mapping of a topological space $Y$ into a product space $X = \prod_{i \in I} X_i$ . Then $f$ is continuous at a point $a \in Y$ if and only if $f_i$ is continuous at $a$ for each $i \in I$ .

Since $f_i = \mathrm{pr}_i \circ f$ , this is just a particular case of Proposition 4 of § 2, no. 3.

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

Cited by

Proposition 5, § 10
§ 4. Product of topological spaces
Corollary 1, § 4
Corollary 2, § 4
Proposition 2, § 8
Theorem 2, § 3
Theorem 2, § 4
Proposition 25, § 2
Remark, § 6
Proposition 1, § 6
Proposition 5, § 1

Cites

Proposition 4 of § 2, no. 3