Metric Space is Hausdorff

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

Then $M$ is a Hausdorff space.

Proof
Let $x, y \in A: x \ne y$.

Then from Distinct Points in Metric Space have Disjoint Open Balls, there exist open $\epsilon$-balls $B_\epsilon \left({x}\right)$ and $B_\epsilon \left({y}\right)$ which are disjoint open sets containing $x$ and $y$ respectively.

Hence the result by the definition of Hausdorff space.