Hereditarily collectionwise normal 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

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:

1. every subspace of it is collectionwise normal
2. 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

• Metrizable space
• Linearly orderable space
• Elastic space
• Monotonically normal space

Weaker properties

• Collectionwise normal space
• Hereditarily normal space
• Normal space
• Collectionwise Hausdorff space

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