# continuous surjection from $GL_2(\Bbb R)$ to closed unit disc in the complex plane

Does there exist a continuous surjective map from $$G = GL_2(\Bbb R)$$ to the closed unit disc $$\{z \in \Bbb C: |z| \le 1\}$$ in $$\Bbb C$$?

I know that $$C^{*}$$ sits inside $$G$$ as a subgroup. But it didn't help me. So kindly share some thoughts. Thank you.

Yes. Let $$A=\pmatrix{a&b\\ c&d}$$ and denote its Frobenius norm $$\sqrt{a^2+b^2+c^2+d^2}$$ by $$\|A\|_F$$. Consider $$f(A)=\frac{\sqrt{a^2+d^2}}{\|A\|_F}\left(\cos\|A\|_F+i\sin\|A\|_F\right).$$ Clearly $$f$$ is a continuous function that maps $$GL_2(\mathbb R)$$ into the closed unit disc on the Argand plane. It is also surjective: for any $$0\le r\le1$$ and $$\theta>0$$, we have $$f(A)=r(\cos\theta+i\sin\theta)$$ when $$A=\frac{1}{\sqrt{2}}\pmatrix{r\theta&-\sqrt{1-r^2}\theta\\ \sqrt{1-r^2}\theta&r\theta}.$$
A similar idea as user1551, but a bit simpler example can be constructed. Let $$f:GL_2(\mathbb{R})\rightarrow \mathbb{C}$$ be defined by $$f(A)=\frac{b^2}{a^2+b^2} e^{id}$$ where $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$