Discrete collection

Definition
A collection of subsets of a topological space is termed discrete if for every point in the whole space, there is an open subset containing that point, that intersects at most one element of the subset.

Clearly, any discrete collection of subsets must be pairwise disjoint. Further, it is also clear that the closures of all members of the collection are pairwise disjoint (because if a point lies in the closure of two subsets, then any open set containing that point must intersect both). However, the closures being pairwise disjoint is not a sufficient condition for discreteness.