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
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
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|hereditarily compact space|
- Hausdorff space: The only Noetherian Hausdorff spaces are finite spaces with the discrete topology.
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.