A connected group which is disconnected

I think this is not an open neighbourhood in $\mathrm{SL} (2,\mathbb{R})$. For example, for each given $\varepsilon >0$, we have the element $$ K=\begin{pmatrix} 1&\sqrt{\varepsilon}\\ \sqrt{\varepsilon}&1+\varepsilon \end{pmatrix} $$ that satisfies $\Vert K-\mathrm{Id}_2\Vert_2\leq\sqrt{3}\varepsilon $ but $K$ does not lie in the neighbourhood described in your question.

It turns out that $AN$ is not open. Your error lies in the assertion that $O_\varepsilon$ is an open subset of $SL(2,\Bbb R)$. It is not.