Query regarding a compact set in a open set $\Omega$ in $\Bbb{C}$

Question

Suppose $\Omega$ is a bounded open set in $\Bbb{C}$
For any $\delta>0$ let us define a new set $\Omega_\delta=\{z\in\Omega|\overline{D_\delta(z)}\subset\Omega\}$.
Is $\Omega_\delta$ compact?
Compactness$\iff$Closed and bounded
$\Omega_\delta\subset \Omega\implies\Omega_\delta$ is bounded.
Is it closed? Take $z$ be limit point of $\Omega_\delta$. Then $D_\delta(z)\setminus\{z\}\cap\Omega_\delta\ne\emptyset\implies z_0\in D_\delta(z)\setminus\{z\}\cap\Omega_\delta$(say).
I can't proceed further. How to solve the problem?

Answer 1

The set $\Omega_\delta$ doesn't have to be compact. Suppose that $\Omega=D_2(0)$ and that $\delta=1$. Then$$(\forall n\in\mathbb N):1-\frac1n\in\Omega_\delta.$$However, $1\notin\Omega_\delta$, since $2\in\overline{D_1(1)}$. But $1=\lim_{n\to\infty}1-\frac1n$. This proves that, in this case, $\Omega_\delta$ is not closed.