Open sets intersecting on boundary Let $A,B$ be open and connected subsets of a simply connected domain $Ω\subset \mathbb{C}$, such that
$$\partial Ω\cap \partial A\neq\emptyset,\partial Ω\cap \partial B\neq\emptyset, \partial A\cap \partial B\neq\emptyset$$ 
and the set
$$(\partial Ω\cap \partial A)\cup (\partial Ω\cap \partial B)$$
is a connected subset of $\partial Ω$. Does this imply that
$$A\cap B\neq \emptyset?$$
 A: Consider the case
\begin{align*}
\Omega &= \mathbb{D}\\
A &= \{z \in \mathbb{D} \mid \operatorname{Im}(z) > 0\}\\
B &= \{z \in \mathbb{D} \mid \operatorname{Im}(z) < 0\}.
\end{align*}
Note that we have
\begin{align*}
\partial\Omega &= S^1\\ 
\partial A &= \{e^{i\theta} \mid 0 \leq \theta \leq \pi\}\cup[-1, 1]\\ 
\partial B &= \{e^{i\theta} \mid \pi \leq \theta \leq 2\pi\}\cup [-1, 1]
\end{align*}
and therefore
\begin{align*}
\partial\Omega\cap\partial A &= \{e^{i\theta} \mid 0 \leq \theta \leq \pi\} \neq \emptyset\\
\partial\Omega\cap\partial B &= \{e^{i\theta} \mid \pi \leq \theta \leq 2\pi\} \neq \emptyset\\
\partial A\cap \partial B &= [-1, 1] \neq \emptyset.
\end{align*}
Furthermore, $(\partial\Omega\cap\partial A)\cup(\partial\Omega\cap\partial B) = S^1$ which is a connected subset of $\partial\Omega$, so all of your conditions are met. However, $A$ and $B$ are disjoint.

In the above example, $\partial\Omega\cap\partial A\cap\partial B = \{-1, 1\}$ is disconnected. If you want an example with $\partial\Omega\cap\partial A\cap\partial B$ connected, consider $\Omega = \mathbb{D}$, $A = \{z \in \mathbb{D} \mid |z - \frac{1}{2}| < \frac{1}{2}\}$, and $B = \Omega\setminus\bar{A}$.
