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.