noetherian topological space

A topological space is noetherian if it satisfies the ascending chain condition for inclusions of open sets. Equivalently, it satisfies the descending chain condition for inclusions of closed sets.

Typical examples are the underlying topological spaces of (classical) quasiprojective varieties over a field kk in Zariski topology. The fact that the affine space 𝔸 k n\mathbb{A}^n_k is noetherian is a consequence of the Hilbert basis theorem.

Every noetherian topological space is a closed union of a finitely many irreducible topological spaces (its irreducible components).