Let X be an infinite set. For $\displaystyle p\in X,q\in X$ define,

$\displaystyle d(p,q)=\left\{\begin{array}{cc}1(p\not=q)\\0(p=q)\ end{array}\right.$

Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

Proving it's metric is easy no problem there.

The rest I just want to make sure I'm thinking of this correctly.

Every subset of the metric space is finite, therefore every set is closed, compact, and not open.

Does this sound right?