Corollary 2

Tag 03QF, permanent. Cite it as [Bourbaki, Tag 03QF].

Theory of Sets, chapter II, § 5. Product of a family of sets

Let $(X_\iota)_{\iota \in I}$ and $(Y_\varkappa)_{\varkappa \in K}$ be two families of sets. Then

\left( \bigcup_{\iota \in I} X_\iota \right) \times \left( \bigcup_{\varkappa \in K} Y_\varkappa \right) = \bigcup_{(\iota, \varkappa) \in I \times K} (X_\iota \times Y_\varkappa)

and, if I and K are non-empty,

\left( \bigcap_{\iota \in I} X_\iota \right) \times \left( \bigcap_{\varkappa \in K} Y_\varkappa \right) = \bigcap_{(\iota, \varkappa) \in I \times K} (X_\iota \times Y_\varkappa).

The proof follows the pattern of the proof of the Corollary to Proposition 8.

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

Nothing.

Cites

Corollary to Proposition 8