Noetherian space

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

Definition

Symbol-free definition

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

• 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).
• 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.
• It is expressible as a union of finitely many irreducible closed subspaces, none of which is properly contained in another.

Definition with symbols

A topological space $X$ is termed Noetherian if given any descending chain of closed subsets:

$C_{1}\supseteq C_{2}\supseteq C_{3}\supseteq \ldots$

there exists a $n$ such that $C_{n}=C_{n+1}=\ldots$.

Relation with other properties

Stronger properties

• Irreducible space

Weaker properties

• 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.