I am currently reading Fuchsian Groups by Svetlana Katok. The following proposition appears on page 30:
There is a misprint. It is supposed to be $z_0$ not $w_0$.
The next paragraph shows that $\text{E}$ is closed.
I am not able understand why $\beta : \text{SL}(2, \mathbb{R}) \rightarrow \mathcal{H} $ must be a continuous map.
This assertion is precisely that $z_0 \mapsto \frac{az_0 + b}{cz_0 + d}$ is continuous in $\{(a,b,c,d) \in \mathbb{R}^4 \mid ad - bc = 1 \}$ for any given fixed $z_0 \in \mathbb{H}$. The restriction $ad - bc = 1$ implies that it is impossible for both $c, d$ to be $0$. One sees from the definition of $\mathbb{H}$ that for any $(c,d)$ with $c \neq 0$, that $\text{Im}(cz_0 + d) \neq 0$, and that for any pair of the form $(0,d)$ with $d \neq 0$, then $cz_0 + d = d \neq 0$. So the denominators of these rational functions never vanish, and well-definition/continuity is elementary.