Is there a metric on the reals $\Bbb R$ so that a subset of $\Bbb R$ is open iff its complement is finite? I wanted to know if there is a distance function $d$ on $\Bbb R$ so that a nonempty subset $U$ of $\Bbb R$ is open with respect to $d$ if and only if its complement $\Bbb R$ \ U is finite ?

No, this is not possible.  The resulting topology is not Hausdorff, and any metric topology must be.
