## hereditarily normal, mutually separated subsets

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 $H \cap \overline{K}$ $=\overline{H} \cap K =\emptyset$