# 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$: Regular and Normal Spaces