Intersection of an unbounded domain and annulus around a boundary point (with controlled radius) is a domain

Question

Is it true that given any unbounded domain $D$ strictly in $\mathbb C$ and a boundary point $z_0$, and two real numbers $R > r > 0$, I can choose some $r' < r$ and $R' > R$ such that $\{ z \in D : r' < |z - z_0| < R'\}$ is a domain?

Here an open subset of $\mathbb C$ is called a domain if any two points in the subset can be joined by gluing straight broken line segments in the subset.

Answer 1

No. Let $$D = \mathbb C \setminus \{ z = x + iy \mid x \in \{0\} \cup \{1/n \mid n \in \mathbb N \}, y \ge 0 \} .$$ Then $z_0 = 0$ is a boundary point of $D$. Now let $A$ be any annulus around $0$. Then $A \cap D$ has infinitely many connected components.