Discrete Space is Fully Normal
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Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is fully normal.
Proof
We have that a Discrete Space is fully $T_4$.
Then we note that from Discrete Space satisfies all Separation Properties, a discrete space is a $T_1$ (Fréchet) space.
Therefore, by definition, $T$ is fully normal.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $6$