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 that is collectionwise Hausdorff is also Hausdorff.

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