The german Wikipedia page states that the inversion $\frac{1}{z}$ is a biholomorphism from the unit disk $ \mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$ to $\mathbb{C} \backslash \overline{\mathbb{D}} $ where $\overline{\mathbb{D}}$ denotes the closed unit disk. https://de.wikipedia.org/wiki/Biholomorphe_Abbildung
First of all, I am confused about $z=0$, because it would imply (on the Riemann sphere) $1/0 = \infty $ where $\infty \notin \mathbb{C}$. My Intuition tells me that this is not possible since $\mathbb{D}$ is simply connected whereas $\mathbb{C} \backslash \overline{\mathbb{D}}$ is not.
Can we state then $f: \mathbb{D}\backslash \{0\} \to \mathbb{C} \backslash \overline{\mathbb{D}} $ is a Biholomorphism? And in general: Let U be an open, connected and simply connected space $U \subset \mathbb{C}$ (such as $\mathbb{D}$) and P an open twice connected (?) space in $\mathbb{C}$. Does there exist a biholomorphism between U and P ? The Riemann Mapping theorem states that connected and simply connected open Subsets of $\mathbb{C}$ are Biholomorphic, but it doesn't state that there is no possible Biholomorphism for the above example.