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:

a perfectly $T_4$ space
a $T_1$ (Fréchet) space.

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