I have a question regarding the Heisenberg group. I understand that this group is simply related. In the matrix case, the argument to verify this is the following.
The application $(a,b,c)\mapsto \begin{bmatrix} 1 & a & b\\ 0&1&c\\ 0&0&1 \end{bmatrix},\, a,\, b,\, c\in\mathbb{R}$ is a homeomorphism.Therefore the Heisenberg group is simply connected because $\mathbb{R}^3$ is simply connected.
Question 1. The set $H=\left\{\mathrm{e}^{a\partial_x+bx+cI}:a,b,c\in\mathbb{R}\right\}$ is the Heisenberg group? (with operators)
Similarly to the matrix case.\
Question 2. The application $(a,b,c)\mapsto \mathrm{e}^{a\partial_x+bx+cI}$ is a homeomorphism?