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

Symbol-free definition

A topological space is said to be collectionwise Hausdorff if it is T1 and given any discrete closed subset (viz a closed subset that is discrete 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.

Definition with symbols

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Relation with other properties

Stronger properties

• Metrizable space
• Collectionwise normal 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