Noetherian space
From Encyclopedia of Mathematics
A topological space 
in which every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets of 
ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space 
has a finite covering by Noetherian subspaces, then 
is itself Noetherian. A space 
is Noetherian if and only if every open subset of 
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 
the space 
(the spectrum of 
) is Noetherian if and only if 
is a Noetherian ring, where 
is the nil radical of 
.
References
| [1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001 |
Comments
References
| [a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
How to Cite This Entry:
Noetherian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_space&oldid=23912
Noetherian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_space&oldid=23912
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article