Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points. A closed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most one point from S. Every T1 space which is collectionwise Hausdorff is also Hausdorff.

Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.