Definition:Noetherian Topological Space
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Definition
Definition 1
A topological space $T = \struct {S, \tau}$ is Noetherian if and only if its set of closed sets, ordered by the subset relation, satisfies the descending chain condition.
Definition 2
A topological space $T = \struct {S, \tau}$ is Noetherian if and only if its set of open sets, ordered by the subset relation, satisfies the ascending chain condition.
Definition 3
A topological space $T = \struct {S, \tau}$ is Noetherian if and only if each non-empty set of closed sets has a minimal element with respect to the subset relation.
Definition 4
A topological space $T = \struct {S, \tau}$ is Noetherian if and only if each non-empty set of open sets has a maximal element with respect to the subset relation.
Also see
- Results about Noetherian topological spaces can be found here.
Source of Name
This entry was named for Emmy Noether.