I haven't been able to find a proper example to solve item c. Item b is shown so that the context is clear.
b. If $A$ is closed, $B$ is compact, and $A \cap B=\emptyset$ , prove that there is $d > 0$ such that $|x-y|\geq d$ for all $y\in A$ and $x\in B$.
c. Give a counterexample in $\Bbb R^2$ if $A$ and $B$ are closed but neither is compact.
As a hint, think about what you lose by assuming the sets are closed but neither is compact (that is, think about how we characterize compact sets in $R^2$). Both sets lack something that you can use to your advantage.