# Definition:Noetherian Topological Space

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## 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.

## Also see

## Source of Name

This entry was named for Emmy Noether.