# Definition:Noetherian Topological Space

## Definition

### Definition 1

A topological space $T = \left({S, \tau}\right)$ is Noetherian if and only if its set of closed sets, ordered by inclusion, satisfies the descending chain condition.

### Definition 2

A topological space $T = \left({S, \tau}\right)$ is Noetherian if and only if its set of open sets, ordered by inclusion, satisfies the ascending chain condition.

## Source of Name

This entry was named for Emmy Noether.