Hereditarily collectionwise normal space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed '''hereditarily collectionwise normal''' if every subspace of it is [[collectionwise normal space|collectionwise normal]]. | A [[topological space]] is termed '''hereditarily collectionwise normal''' or '''completely collectionwise normal''' if it satisfies the following two conditions: | ||
# every subspace of it is [[collectionwise normal space|collectionwise normal]] | |||
# every almost discrete collection of closed subsets can be separated by pairwise disjoint open subsets (here, ''almost discrete'' means discrete in the union). | |||
==Formalisms== | ==Formalisms== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Metrizable space]] | * [[Weaker than::Metrizable space]] | ||
* [[Linearly orderable space]] | * [[Weaker than::Linearly orderable space]] | ||
* [[Elastic space]] | * [[Weaker than::Elastic space]] | ||
* [[Monotonically normal space]] | * [[Weaker than::Monotonically normal space]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Collectionwise normal space]] | * [[Stronger than::Collectionwise normal space]] | ||
* [[Hereditarily normal space]] | * [[Stronger than::Hereditarily normal space]] | ||
* [[Normal space]] | * [[Stronger than::Normal space]] | ||
* [[Stronger than::Collectionwise Hausdorff space]] | |||
==Metaproperties== | ==Metaproperties== | ||
{{subspace-closed}} | {{subspace-closed}} | ||
Latest revision as of 00:15, 25 October 2009
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
This is a variation of normality. View other variations of normality
Definition
A topological space is termed hereditarily collectionwise normal or completely collectionwise normal if it satisfies the following two conditions:
- every subspace of it is collectionwise normal
- every almost discrete collection of closed subsets can be separated by pairwise disjoint open subsets (here, almost discrete means discrete in the union).
Formalisms
In terms of the hereditarily operator
This property is obtained by applying the hereditarily operator to the property: collectionwise normal space
Relation with other properties
Stronger properties
Weaker properties
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