While going through some research papers, I came across a result of D. Bucur where the existence of a minimizer for the general $k^{th}$ eigenvalue of the Dirichlet Laplacian among a class of quasi-open sets with fixed measure was shown.
I have looked into the definition of quasi-open sets from A. Henrot, Extremum problems for eigenvalues of elliptic operator. But I was unable to find any insight as to what kind of sets one might expect to be quasi-open.
First, in Definition 2.4.3, they defined the capacity of a set as
Let $D$ be a bounded open set in $\mathbb{R}^n$. For any compact subset $K$ in $D$, define $$\text{cap}_D(K)= \inf \left\{\int_D|\nabla v|^2: v\in C_0^{\infty}, v\geq 1 \text{ surrounding } K \right\}$$
The definition can be extended for an open subset $\omega$ of $D$ as
$$\text{cap}_D(\omega):= \sup \{\text{cap}_D(K): K \text{ compact, } K\subset \omega\}$$
Quasi-open sets as in Definition 2.4.4 is defined as follows,
A subset $\Omega$ of $D$ (a bounded open set in $\mathbb{R}^n$) is quasi-open if there exist a decreasing sequence of open sets $\omega_n$ such that $$\displaystyle \lim_{n\to +\infty} \text{cap}_D(\omega_n)=0$$ where $\Omega \cup \omega_n$ is open for all $n$ and $\text{cap}_D(\omega_n)$ is the capacity of $\omega_n$ relative to $D$.
I would love to get an insight on how the capacity of a set fits in the definition of quasi-open sets. Also, it would be helpful if I can get a detailed example of a quasi-open set or a reference to find one.
I would also appreciate if someone can provide me with some intuition as to how to look at the idea of capacity of a set and which aspect of a set does it actually measure.