Definition 1

A topological space $X$ is said to be normal if it is Hausdorff and satisfies the following axiom:

$(O_v)$ If $A$ and $B$ are any two disjoint closed subsets of $X$, there exists a continuous mapping of $X$ into $[0, 1]$ which is equal to $0$ at every point of $A$ and to $1$ at every point of $B$.

Clearly every normal space is completely regular; but there are completely regular spaces which are not normal (see Exercises 9, 10, 13, 26 and § 5, Exercises 15 and 16).

The statement of Axiom $(O_v)$, like that of $(O_{IV})$, involves the real line $\mathbf{R}$ as an auxiliary set. But there is a condition equivalent to $(O_v)$ which involves no auxiliary set: