Proposition 4

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

Theory of Sets, chapter III, § 6. Infinite sets

Let $f$ be a mapping of a set E onto an infinite set F such that, for each $y \in \mathrm{F}$ , $\overset{-1}{f}(y)$ is countable. Then F is equipotent to E.

For the sets $\overset{-1}{f}(y)$ $(y \in \mathrm{F})$ form a partition of E; hence

\mathrm{Card}(\mathrm{E}) \leqslant \mathrm{Card}(\mathrm{F})\mathrm{Card}(\mathbf{N}) = \mathrm{Card}(\mathrm{F}),

and $\mathrm{Card}(\mathrm{F}) \leqslant \mathrm{Card}(\mathrm{E})$ by Proposition 3 of § 3, no. 2.

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

Proposition 3 of § 3, no. 2