Completely Normal Space is Normal Space
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Theorem
Let $\struct {S, \tau}$ be a completely normal space.
Then $\struct {S, \tau}$ is also a normal space.
Proof
Let $\struct {S, \tau}$ be a completely normal space.
From the definition, $\struct {S, \tau}$ is a completely normal space if and only if:
- $\struct {S, \tau}$ is a $T_5$ space
- $\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
We have that a $T_5$ space is a $T_4$ space.
So:
- $\struct {S, \tau}$ is a $T_4$ space
- $\struct {S, \tau}$ 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 $2$: Separation Axioms: Regular and Normal Spaces