How to show that $\mathbb{R} \times (0,+\infty) \times \mathbb{R}$ is a universal cover of $\mathrm{SL}_2(\mathbb{R})$ by Iwasawa Decomposition?
My attempt By Iwasawa Decomposition, for any $A \in \mathrm{SL}_2(\mathbb{R})$, we have $$ A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} r & 0 \\ 0 & \frac 1r \end{pmatrix} \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}. $$ Thus we get a map $\varphi: \mathbb{R} \times (0,+\infty) \times \mathbb{R} \to \mathrm{SL}_2(\mathbb{R})$. I think it's sufficient to show that $\mathbb{R}$ is a universal cover of the subspace $$ \left\{ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} : \theta \in \mathbb{R} \right\}. $$ But I don't know how to show it rigorously. Can you give me some hints?