Metric Space is Perfectly Normal
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Then $M$ is a perfectly normal space.
Proof
By definition, a topological space is perfectly normal space if and only if it is:
We have that:
- a Metric Space is Perfectly $T_4$
- a Metric Space is $T_2$ (Hausdorff)
- a $T_2$ (Hausdorff) Space is a $T_1$ (Fréchet) 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 $5$: Metric Spaces