Collectionwise Hausdorff 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 Hausdorffness. View other variations of Hausdorffness

Definition

A topological space is said to be collectionwise Hausdorff if it satisfies the following: it is T1 and given any discrete closed subset (viz a closed subset that is a discrete space under the induced topology), we can find a disjoint family of open sets, with each point of the discrete subset contained in exactly one member open set.

Relation with other properties

Stronger properties

• Metrizable space

Weaker properties

• Hausdorff space: For proof of the implication, refer collectionwise Hausdorff implies Hausdorff and for proof of its strictness (i.e. the reverse implication being false) refer Hausdorff not implies collectionwise Hausdorff