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 $k$ in Zariski topology. The fact that the affine space $\mathbb{A}^n_k$ 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).