Noetherian topological space

In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that "every" subset is compact.
Definition.
A topological space formula_1 is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence 
of closed subsets formula_3 of formula_1, there is an integer formula_5 such that formula_6 
From algebraic geometry.
Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant.
A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings. This class of examples therefore also explains the name. 
If "R" is a commutative Noetherian ring, then Spec("R"), the prime spectrum of "R", is a Noetherian topological space. More generally, a Noetherian scheme is a Noetherian topological space. The converse does not hold, since Spec("R") of a one-dimensional valuation domain "R" consists of exactly two points and therefore is Noetherian, but there are examples of such rings which are not Noetherian.
Example.
The space formula_14 (affine formula_15-space over a field formula_16) under the Zariski topology is an example of a Noetherian topological space. By properties of the ideal of a subset of formula_14, we know that if
is a descending chain of Zariski-closed subsets, then
is an ascending chain of ideals of formula_20 Since formula_21 is a Noetherian ring, there exists an integer formula_5 such that 
Since formula_24 is the closure of "Y" for all "Y", formula_25 for all formula_26 Hence