# How to prove this group is homeomorphic to $\mathbb{C}\times \mathbb{C}^*$

I am reading the Fulton& Harris "Representation Theory:A First Course".In Page 136 they gave a group $$G_0 = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}: a \neq 0 \right\} \subset GL_2(\mathbb{C})$$ and they said "topologically this group is homeomorphic to $$\mathbb{C}\times \mathbb{C}^*$$", could I ask that how to prove this? Thanks!

New contributor
Snow is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• Can you think of a continuous map from $G_0$ to $\mathbb{C} x \mathbb{C}^\times$? May 14 at 3:36
• I mean, $a$ and $b$ are right there, and one of them isn't allowed to be zero... May 14 at 4:10
• But I thought as @StefanDawydiak said, this homeomorphism needed to be proved by constructing a continuous map, or maybe prove the fundamental group of $G_0$ is isomorphic as the fundamental group of $\mathbb{C}\times \mathbb{C}^*$...
– Snow
May 14 at 7:08
• Proving the fundamental groups are isomorphic is not enough, but this question is much simpler than that. I think you are overthinking this. Elements of $G_0$ are given by two complex numbers, one nonzero. Where should the map you want to construct send these two numbers? As Jonathan says, this map has a very simple definition. May 14 at 11:54
• I see. Thank you very much!
– Snow
May 15 at 5:15