Proposition 9

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

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

If $(\mathrm{E}_\alpha^\lambda, f_{\beta\alpha}^{\mu\lambda})$ is a direct system of sets relative to a product $\mathrm{I} \times \mathrm{L}$ of two directed sets, then (up to a canonical bijection) we have

(32)

\varinjlim_{\alpha,\ \lambda} \mathrm{E}_\alpha^\lambda = \varinjlim_{\lambda}\ (\varinjlim_{\alpha} \mathrm{E}_\alpha^\lambda).

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.

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Cites

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