Completely Normal Space is Normal Space

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Theorem

Let $\left({S, \tau}\right)$ be a completely normal space.


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


Proof

Let $\left({S, \tau}\right)$ be a completely normal space.

From the definition, $\left({S, \tau}\right)$ is a completely normal space iff:

$\left({S, \tau}\right)$ is a $T_5$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.


We have that a $T_5$ space is a $T_4$ space.

So:

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

which is precisely the definition of a normal space.

$\blacksquare$


Sources