I was solving this problem:
Let $D=B(0,1)\setminus\{0\}$ and $B=B(0,1)\setminus \overline{B(0,\frac{1}{2})}.$ There is an biholomorphic function $f:D\to B?$
And the answer is no. This is easy to see using the Riemann principle we extend $f$ in $0$ and contradict the injectivity of $f$ using two disjoint opens set in the domain.
After that I was thinking if there are any other open set $U$ in complex plane such that there is a biholomorpginc function $f:B\to U.$
For exemple if $U=B(0,r)\setminus\{0\}$ then is easy to find an biholomorfism.
My first question, $U$ can be unbounded?
My second question, I true that if I take a simply connected open set from plane say $W$ and i remove a point say $w$ then $W\setminus\{w\}$ is the only exemple for such a $U$?