Distance between two sets ($d(A,B)={\inf }_{a\in A,b\in B}d(a,b)$)

The set distance is the infimum of pairwise metric distances. Two nonempty disjoint compact sets have positive set distance: continuity attains the minimum on their Cartesian product, and a zero minimum would give a common point. Disjoint closed sets without compactness need not have positive separation.