# 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.