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

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

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

Background:

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