Noetherian space

2020 Mathematics Subject Classification: Primary: 54F65 [MSN][ZBL]

A topological space $X$ which satisfies the descending chain condition for closed subspaces: every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space $X$ has a finite covering by Noetherian subspaces, then $X$ is itself Noetherian. A space $X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space is the union of finitely many irreducible components.

Examples of Noetherian spaces are some spectra of commutative rings (cf. Spectrum of a ring). For a ring $A$ the space $\mathrm{Spec}(A)$ (the spectrum of $A$) is Noetherian if and only if $A/J$ is a Noetherian ring, where $J$ is the nil radical of $A$.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001