Perfectly Normal Space is Completely Normal Space

Theorem
Let $\left({X, \tau}\right)$ be a perfectly normal space.

Then $\left({X, \tau}\right)$ is also a completely normal space.

Proof
Let $T = \left({X, \tau}\right)$ be a perfectly normal space.

From the definition:


 * $\left({X, \tau}\right)$ is a perfectly $T_4$ space
 * $\left({X, \tau}\right)$ is a $T_1$ (Fréchet) space.