I am reading by myself Hörmander's book An Introduction to Complex Analysis and I have some doubts on the paragraphs after Runge's Theorem:
Let $\Omega$ an open subset of $\mathbb{C}$ and $K\subset\Omega$ a compact set. On page 8 of the book, 8 lines above Theorem 1.3.3. he says
For every compact set $K\subset \Omega$ then the hull $\hat{K_{\Omega}}$ is thus a compact subset of $\Omega$ containing $K$.
However, on Christine Laurent-Thiébaut's book Holomorphic Function Theory in Several Variables page 116 there is a remark. We have seen that $\hat{K_{\Omega}}$ is always bounded and closed in $\Omega$, but it is not in general a compact set in $\Omega$. See the print.
I understood that it is bounded and closed on $\Omega$ why it is not compact? That got me confused. Do you have some example?
And there is a part of the following theorem thah I do not understand.
Theorem 1.3.3. $\hat{K_{\Omega}}$ Is the union of $K$ and the components of $\Omega\setminus K$ which are relatively compact on $\Omega$.
On the proof he denotes by $K_1$ the union of $K$ and all such components. Now $\Omega\setminus K_1$ is open, since is a union of open components of $\Omega\setminus K$. Hence is $K_1$ is compact. I do not understand why $K_1$ is compact.
Than you!