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.