Metric Space is Completely Normal

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is a completely normal space.

Proof
By definition, a topological space is completely normal space  it is:
 * a $T_5$ space
 * a $T_1$ (Fréchet) space.

We have that:
 * a Metric Space is $T_5$
 * a Metric Space is $T_2$ (Hausdorff)
 * a $T_2$ (Hausdorff) Space is a $T_1$ (Fréchet) Space.