Fully Normal Space is Normal Space

Theorem

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

Then $T$ is a normal space.


Proof

From the definition, $T$ is fully normal if and only if:

$T$ is fully $T_4$
$T$ is a $T_1$ (Fréchet) space.

We have that a fully $T_4$ space is also a $T_4$ space.

So:

$T$ is a $T_4$ Space
$T$ is a $T_1$ (Fréchet) space

which is precisely the definition of a normal space.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
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