Show that the Riemann sphere complement the unit disc is a hyperbolic Riemann surface

Question

I am trying to understand what $\hat{\mathbb{C}} \backslash \bar{\mathbb{D}}$ in the area of Riemann surfaces.

$\hat{\mathbb{C}}$ is the Riemann sphere and $\bar {\mathbb{D}}$ is the closure of the unit disk.

1. Is $\hat{\mathbb{C}} \backslash \bar{\mathbb{D}} = \{ z \in \mathbb{C} : |z| > 1 \} \cup \{\infty \}$?

2. I think this is a non-simply connected space because there is a hole in the middle of it and I am guessing that I will not be able to find a homotopy for this space. If so, how do I show that this is the case?

3. Is this a hyperbolic Riemann surface?

For 2, from what I have learnt one method to show that this is a hyperbolic Riemann surface is by looking at its universal covering, and another is by showing that it is biholomorphic to another hyperbolic surface, but I am not sure how to use either of these methods here.

P.S. This is my first time asking a question here. Thanks!

Answer 1

The answer to 1 and 3 is yes. Furthermore the formula that lies behind this answer shows that there is no "hole in the middle of it" and that the answer to 2 is no. All these answers are obtained using the inversion formula $$f(z) = \begin{cases} \frac{1}{z} & \quad\text{if $z \not\in \{0,\infty\}$} \\ \infty & \quad\text{if $z=0$} \\ 0 & \quad\text{if $z = \infty$} \end{cases} $$ which maps $\hat{\mathbb C} \setminus \bar{\mathbb D}$ to $\mathbb D$ biholomorphically.