Noetherian space: Difference between revisions

Latest revision as of 22:26, 15 November 2015

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

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 $X$ 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 $C_{1}\supseteq C_{2}\supseteq C_{3}\supseteq \ldots$ there exists a $n$ such that $C_{n}=C_{n+1}=\ldots$ .
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.

@@ Line 10: / Line 10: @@
 | 1 || descending chain of closed subsets || Any descending chain of [[closed subset]]s 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 <math>C_1 \supseteq C_2 \supseteq C_3 \supseteq \ldots</math> there exists a <math>n</math> such that <math>C_n = C_{n+1} = \ldots</math>.
 |-
-| 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. ||
+| 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 subset]]s 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. a closed subset which does not strictly contain any other member of the collection.||
+| 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. ||
 |}