Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space $$X$$ is said to be collectionwise Hausdorff if given any closed discrete subset of $$X$$, 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 $$S\subseteq X$$ being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of $$S$$ are isolated in $$S$$).

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 $$S$$ of $$X$$, every singleton $$\{s\}$$ $$(s\in S)$$ is closed in $$X$$ and the family of such singletons is a discrete family in $$X$$.)


 * Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.