# 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.