A region $G \subset \mathbb{C}$ is called $n$-connected if $\mathbb{C}_{\infty} \backslash G$ has $n+1$ components (Conway chapter 15). Consider now two regions: an annulus $A=\{r<|z|<1\}$ and an unbounded region $B$ with two holes $B = \mathbb{C} \backslash (D_1 \cup D_2)$, where $D_1, D_2$ are some disjoint closed disks.
1) $\mathbb{C}_{\infty} \backslash A = \{|z|\leq r\} \cup \{|z| \geq 1\} \cup \{\infty\}$ has two components because $\{|z| \geq 1\} \cup \{\infty\}$ is connected.
2) $\mathbb{C}_{\infty} \backslash B = D_1 \cup D_2 \cup \{\infty\}$ has $3$ components.
But $A$ and $B$ are conformally equivalent, so the connectivity should be the same. Where is my mistake?