Definition:Noetherian Topological Space

Definition
Let $X$ be a topological space.

Definition 1
The space $X$ is Noetherian its set of closed subsets, ordered by inclusion satisfies the descending chain condition.

Definition 2
The space $X$ is Noetherian its set of open subsets, ordered by inclusion satisfies the ascending chain condition.

Also see

 * Equivalence of Definitions of Noetherian Topological Space