Corollary

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

Theory of Sets, chapter III, § 7. Inverse limits and direct limits

If $u_\alpha$ is injective (resp. bijective) for each $\alpha \in \mathrm{I}$ , then $u$ is injective (resp. bijective).

With the notation of Proposition 2, the images $u_\alpha(\mathrm{E}_\alpha)$ also form an inverse system of subsets of the $\mathrm{E}'_\alpha$ , and we have

(9)

u(\mathrm{E}) \subset \varprojlim u_\alpha(\mathrm{E}_\alpha);

but the two sides of this relation are not necessarily equal (Exercise 4).

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

Exercise 4
Proposition 2