Lemma 5

Let K be a compact subset of $\mathbf{C}$. We denote by $\widehat{K}$ the union of K and of the connected components of $\mathbf{C}-$ K which are relatively compact. Then the set $\widehat{K}$ is compact.

Since K is compact, there exists a real number $r >0$ such that K is contained in the open disk D with center 0 and radius $r$. Then $\mathbf{C}-$ K contains $\mathbf{C}-$D. Since the space $\mathbf{C}-$ D is connected (being homeomorphic to $[r,+\infty [\times \mathbf{S}^1)$, it is contained in a connected component U of $\mathbf{C}^-$K. Every other connected component of $\mathbf{C}^-$ K is contained in D, hence is bounded. The connected component U is therefore the unique unbounded connected component of $\mathbf{C}^-$ K, that is, $U =\mathbf{C}-\widehat{K}$. Since U is open and contains the complement of the disk D, the set $\widehat{K}$ is compact.