metric spaces are Hausdorff

Suppose we have a space $X$ and a metric $d$ on $X$. We’d like to show that the metric topology that $d$ gives $X$ is Hausdorff.

Say we’ve got distinct $x,y\in X$. Since $d$ is a metric, $d(x,y)\ne 0$. Then the open balls $B_x=B(x,\frac{d(x,y)}{2})$ and $B_y=B(y,\frac{d(x,y)}{2})$ are open sets in the metric topology which contain $x$ and $y$ respectively. If we could show $B_x$ and $B_y$ are disjoint, we’d have shown that $X$ is Hausdorff.

We’d like to show that an arbitrary point $z$ can’t be in both $B_x$ and $B_y$. Suppose there is a $z$ in both, and we’ll derive a contradiction. Since $z$ is in these open balls, and . But then , contradicting the triangle inequality.

So $B_x$ and $B_y$ are disjoint, and $X$ is Hausdorff.$\mathrm{\square }$