If $D_1, D_2$ are unbounded, is $D_1\cap D_2$ unbounded? Let $D_1$ and $D_2$ be two unbounded Cauchy domains in $\mathbb C$ with $D_1\cap D_2 \neq \emptyset$.
If $D= D_1\cap D_2$, then is $D$ also necessarily unbounded? I am thinking yes, but I am not too sure.
Edit:

Definition: (Cauchy Domain)
A subset $A$ of $\mathbb C$ is said to be a Cauchy domain if:

*

*$A$ is open

*$A$ has a finite number of components, the closure of any two which are disjoint

*The boundary of $A$ is composed of a finite positive number of closed rectifiable Jordan curves, no two of which intersect.



Yes
Identify C with the sphere. The complement of any unbounded Cauchy domain will be compact. As such every unbounded Cauchy domain will be an open neighborhood of infinity. The intersection of two open sets is open. Thus the intersection must be a neighborhood of infinity. So it must be unbounded.
