Definition 3
Tag 049K, permanent. Cite it as [Bourbaki, Tag 049K].
Commutative Algebra, chapter II, § 4. Spectra of rings and supports of modules
A topological space is called Noetherian if every non-empty set of closed subsets of , ordered by inclusion, has a minimal element.
It amounts to the same to say that every non-empty set of open subsets of , ordered by inclusion, has a maximal element, or that every decreasing (resp. increasing) sequence of closed (resp. open) sets is stationary (Set Theory, Chapter 111, § 6, no. 5, Proposition 6).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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