# Noetherian space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

## Contents

## Definition

A topological space is termed **Noetherian** if it satisfies the following equivalent conditions:

No. | Shorthand | A topological space is termed Noetherian if ... | A topological space is termed Noetherian if ... |
---|---|---|---|

1 | descending chain of closed subsets | Any descending chain of closed subsets stabilizes after finitely many steps (in other words, the topological space satisfies the descending chain condition on closed subsets). | for any descending chain of closed subsets there exists a such that . |

2 | minimal element in collection of closed subsets | Any nonempty collection of closed subsets has a minimal element i.e., a closed subset which does not strictly contain any other member of the collection. | |

3 | ascending chain of open subsets | Any ascending chain of open subsets stabilizes after finitely many steps (in other words, the topological space satisfies the ascending chain condition on open subsets). | |

4 | maximal element in collection of open subsets | Any nonempty collection of open subsets has a maximal element i.e. an open subset that is not contained in any other member of the collection. |

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

hereditarily compact space | ||||

compact space |

### Opposite properties

- Hausdorff space: The only Noetherian Hausdorff spaces are finite spaces with the discrete topology.

## Metaproperties

### Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.

View other subspace-hereditary properties of topological spaces

Any subspace of a Noetherian space is Noetherian.