Proposition 4

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

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

If $(\mathrm{E}^\lambda_\alpha, f^{\lambda\mu}_{\alpha\beta})$ is an inverse system of sets relative to a product $\mathrm{I} \times \mathrm{L}$ of preordered sets, then (up to a canonical bijection) we have

(16) \qquad \varprojlim_{\alpha,\lambda} \mathrm{E}^\lambda_\alpha = \varprojlim_\lambda \; (\varprojlim_\alpha \mathrm{E}^\lambda_\alpha).

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

Corollary 1, § 7
Corollary 2, § 7

Cites

Nothing.