Is there a homeomorphism between the closed left-half plane and the closed unit disk of $\mathbb C$

Question

Let $\mathbb D = \{z \in \mathbb C: |z| < 1\}$ and $\mathbb H_l = \{z \in \mathbb C: \text{Re}(z) < 0\}$. I know a particular case of Mobius transformation, $z \mapsto \frac{z+1}{z-1}$, is a diffeomorphism between $\mathbb D$ and $\mathbb H_l$. I am wondering whether there exists a homeomorphism between $\bar{\mathbb D}$ and $\bar{\mathbb H}_l$, i.e., the closed unit disk and the closed left-half plane of $\mathbb C$?

Answer 1

Closed unit disk is compact and closed half plane is not. So there cannot be any homeomorphism between these two.

Answer 2

It is not possible even if you agree for considering $\infty$ point ( and $1/\infty$ = 0, 1/0 = $\infty$ ), since you will have sequence of boundary points of left-half plane ( $ni : n\in \mathbb{N}$ ), which goes to $\infty$, so this would mean that there is a sequence of boundary points of unit disc which goes to zero, which is not true. Without adding $\infty$ it is simply not true, as closed unit disc is compact.