Metric Space is Hausdorff/Proof 1

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Let $M = \struct {A, d}$ be a metric space.

Then $M$ is a Hausdorff space.


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

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

Hence the result by the definition of Hausdorff space.