Noetherian topological space

Summary

A Noetherian topological space is a topological space in which every descending chain of closed subsets eventually stops. It is the topological counterpart of a Noetherian ring: the spectrum of a Noetherian ring with the Zariski topology is a Noetherian space. Every Noetherian space is compact, and so is each of its subspaces. Every Noetherian space is a finite union of irreducible closed subsets, its irreducible components.