Why $\text{SL}_2(\mathbb Z_p)$ is open in $\text{GL}_2(\mathbb Q_p)$?

Question

Let $p$ be a prime number. Denote by $\mathbb Q_p$ and $\mathbb Z_p$ the field of $p$-adic numbers and the ring of $p$-adic integers. Why $\text{SL}_2(\mathbb Z_p)$ is open in $\text{GL}_2(\mathbb Q_p)$ ?

Answer 1

I think it's not open: consider the identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}_p)$ and any basic open neighborhood $$U_n = \left\{ \begin{pmatrix} 1+\varepsilon_1 & \varepsilon_2 \\ \varepsilon_3 & 1+\varepsilon_4 \end{pmatrix} : v_p(\varepsilon_i) \geqslant n \text{ for } i = 1, 2, 3, 4 \right\} \subseteq \operatorname{GL}_2(\mathbb{Q}_p).$$

Then $A_n = \begin{pmatrix} 1 & p^n \\ p^n & 1 \end{pmatrix}$ is in the neighborhood but not in $\operatorname{SL}_2(\mathbb{Z}_p)$, so $U_n \not \subseteq \operatorname{SL}_2(\mathbb{Z}_p)$.