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Noetherian topological space
A topological space $X$ is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence
$Y_{1}\supseteq Y_{2}\supseteq\cdots$ 
of closed subsets $Y_{i}$ of $X$, there is an integer $m$ such that $Y_{m}=Y_{{m+1}}=\cdots$.
As a first example, note that all finite topological spaces are Noetherian.
There is a lot of interplay between the Noetherian condition and compactness:

Every Noetherian topological space is quasicompact.

A Hausdorff topological space $X$ is Noetherian if and only if every subspace of $X$ is compact. (i.e. $X$ is hereditarily compact)
Note that if $R$ is a Noetherian ring, then $\text{Spec}(R)$, the prime spectrum of $R$, is a Noetherian topological space.
Example of a Noetherian topological space:
The space $\mathbb{A}^{n}_{k}$ (affine $n$space over a field $k$) under the Zariski topology is an example of a Noetherian topological space. By properties of the ideal of a subset of $\mathbb{A}^{n}_{k}$, we know that if
$Y_{1}\supseteq Y_{2}\supseteq\cdots$ is a descending chain of Zariskiclosed subsets, then $I(Y_{1})\subseteq I(Y_{2})\subseteq\cdots$ is an ascending chain of ideals of $k[x_{1},\ldots,x_{n}]$.
Since $k[x_{1},\ldots,x_{n}]$ is a Noetherian ring, there exists an integer $m$ such that $I(Y_{m})=I(Y_{{m+1}})=\cdots$. But because we have a onetoone correspondence between radical ideals of $k[x_{1},\ldots,x_{n}]$ and Zariskiclosed sets in $\mathbb{A}^{n}_{k}$, we have $V(I(Y_{i}))=Y_{i}$ for all $i$. Hence $Y_{m}=Y_{{m+1}}=\cdots$ as required.
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Comments
Relationship to noetherian rings
Suppose $R$ is a commutative unital ring. Are the following two concepts equivalent?
* Spec R is a noetherian topological space
* R is noetherian
Re: Relationship to noetherian rings
[Sent this via PM mail as well; sorry about the duplication.]
Apparently not. According to Problem 2.13 (p. 80) from Hartshorne, the latter condition implies the former, but the implication is oneway: Spec(R) can be noetherian even when R is not. I can't supply proofs of these results offhand.