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.

Definition with symbols

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

$C_1\supset C_2\supset C_3\supset \dots$

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

Relation with other properties

Stronger properties

• Irreducible space

Weaker properties

• Compact space