Exercise 14, § 4

Let $K$ be a compact subset of $\mathbf{C}$ and let P (resp. Q) be the closure in $\mathscr{O}(K)$ of the set of germs of polynomial (resp. rational) functions holomorphic in a neighbourhood of K. Now prove that P = Q if, and only if, the set $\mathbf{C}-$ K is connected.