Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.

Here a subset being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of are isolated in ).

Properties

• Every T₁ space that is collectionwise Hausdorff is also Hausdorff.

• Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset of , every singleton is closed in and the family of such singletons is a discrete family in .)

• Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.

Remarks

References