Perfectly Normal Space is Completely Normal Space

Theorem
Let $T = \struct {S, \tau}$ be a perfectly normal space.

Then $T$ is also a completely normal space.

Proof
Let $T = \struct {S, \tau}$ be a perfectly normal space.

From the definition:


 * $T$ is a perfectly $T_4$ space
 * $T$ is a $T_1$ (Fréchet) space.