Corollary 1

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

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

Let $(\mathrm{{E}'}^\lambda_\alpha, {f'}^{\mu\lambda}_{\alpha\beta})$ be another inverse system of sets relative to $\mathrm{I} \times \mathrm{L}$ , and for each $(\alpha, \lambda) \in \mathrm{I} \times \mathrm{L}$ let $u^\lambda_\alpha$ be a mapping of $\mathrm{E}^\lambda_\alpha$ into $\mathrm{{E}'}^\lambda_\alpha$ such that the $u^\lambda_\alpha$ form an inverse system of mappings. Then

(17) \qquad \varprojlim_{\alpha,\lambda} u^\lambda_\alpha = \varprojlim_\lambda \; (\varprojlim_\alpha u^\lambda_\alpha).

The verification is similar to that of Proposition 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

Proposition 4