The existence of "arbitrary large" connected compact sets in the plane Studying some complex analysis I came up with the following hypothesis:

Let $\Omega \subseteq \mathbb C$ be a region (an open and connected set), and let $E \subset \Omega$ be a compact subset. There exists a connected compact set $F$ satisfying $E \subset F \subset \Omega$.

I believe that this is true and I want to prove it. My idea was considering $$F_n=\left\{z \in \Omega:|z| \leq n,\text{dist}(z, \partial\Omega) \geq \frac{1}{n} \right\} $$ for large integers $n$, since $\Omega$ itself is connected and the $F_n$ "tend" to $\Omega$.
Unfortunately, I couldn't prove $F_n$ works.
Is my statement correct? Could you please provide me with a proof?
Thanks!
 A: The sets $F_n$ are compact, but they need not be connected. Consider a region $\Omega$ consisting of the lower half plane, a disk with radius $\frac12$ and centre $k+i$ for all $k \in \mathbb{Z}$, and for each $k\in\mathbb{Z}$ a corridor with width $2^{-(4+\lvert k\rvert)}$ connecting the half plane with the disk. Then the corridors are too narrow, so you have islands of $F_n$ in the disks for large enough $\lvert k\rvert < n$ that aren't connected to the main body in the lower half plane.
The statement is however true (and  from that it follows that $E$ is contained in one connected component of $F_n$ for all large enough $n$).
First, we have $\delta := \operatorname{dist}(E,\,\partial \Omega) > 0$ since $E$ is compact. Thus $E_1 := \{ z : \operatorname{dist}(z,\,E) \leqslant \delta/2\}$ is a compact subset of $\Omega$. Now, $E_1$ is a union of (closed) disks of radius $\delta/2$, hence it can have only finitely many connected components. $\Omega$ is open and connected, hence path-connected. Now you can connect all of the finitely many connected components of $E_1$ with paths in $\Omega$.
