Exercise 3, § 4

If $X$ is a topological space, the following conditions are equivalent:

a) Every subspace of $X$ satisfies axiom $(O'_V)$.

b) Every open subspace of $X$ satisfies axiom $(O'_V)$.

c) Given any two subsets $A$ and $B$ of $X$ such that $A \cap \overline{B} = B \cap \overline{A} = \varnothing$, there exist two disjoint open sets $U, V$ such that $A \subset U$ and $B \subset V$.

A topological space $X$ is said to be completely normal if it is Hausdorff and satisfies these conditions. Every metrizable space is completely normal. A compact space need not be completely normal.