Noetherian topological space [SEP] named after
Noetherian topological space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that "every" subset is compact.
Definition.
A topological space formula_1 is called
Zariski space
In algebraic geometry, a Zariski space, named for Oscar Zariski, has several different meanings:
- A topological space that is Noetherian (every open set is quasicompact)
- A topological space that is Noetherian and also sober (every nonempty closed irreducible subset is the closure of a unique point). The spectrum of any commutative Noetherian ring is a Zariski space in this sense
- A Zariski–Riemann space of valuations of a field