A parallel metrization theorem

Two non-empty sets A, B of a metric space (X, d) are called parallel if d(a,B)=d(A,B)=d(A,b)or any points a is an element of A and b is an element of B. Answering a question posed on mathoverflow.net, we prove that for a cover Cof a metrizable space X by compact subsets, the following conditions are equivalent: (i) the topology of X is generated by a metric d such that any two sets A,B is an element of Care parallel; (ii) the cover C is disjoint, lower semicontinuous and upper semicontinuous.