I want to describe the k-th connected component of $X = \mathbb{C}\setminus (A\cup B)$ where $A = [0,\infty) \mbox{ and } B = \{z = r\mbox{cis}(\theta) \mbox{ | } 0 \leq \theta \leq \infty, r = \theta\}$
What I have is:
Let $k \in \mathbb{N}$
$A_k = [2\pi - 2\pi k, 2\pi k),B_k = \{z = r\mbox{cis}(\theta) \mbox{ | } 2\pi - 2\pi k \leq \theta < 2\pi k, r = \theta\}.$
Let $\mathcal{F}_k$ be an indexed family of substets of $\mathbb{C}$ such that $\mathcal{F}_k = \{ C_{\alpha} \subseteq \mathbb{C} \mbox{ | } \partial C_{\alpha} = A_k \cup B_k \}$
The k-th connected component of $X$ are given by $C_{k} = \bigcup_{\alpha}\{\mathcal{F}_{k}\}\setminus C_{k-1}$
I am correct ?
Thanks!