Show that the following statements are equivalent for any topological space (X, \tau).

(a) Whenever A, B are mutually separated subsets of X, there exist open disjoint U, V such that A \subseteq U and B \subseteq V.

(b) (X, \tau) is hereditarily normal.

Background:

Definition- Sets H and K are mutually separated in a space X if and only if H \cap \overline{K} =\overline{H} \cap K =\emptyset