Fully Normal Space is Normal Space
Jump to navigation
Jump to search
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