A counter example

Question

If a set is compact in $Z(\mathbb{A})\setminus GL(2,\mathbb{A})$,then can it be compact in $GL(2,\mathbb{A})$ ? ($\mathbb{A}$, is the adele ring of $F$ on which $GL(2)$ is and $Z$ is the center of $GL(2)$) Could you suggest a counter example? I thought that i can use the function $GL(2,\mathbb{A}) \mapsto Z(\mathbb{A})\setminus GL(2,\mathbb{A})$ given by $g \mapsto Z(\mathbb{A}) g$: Take the set $\{Z(\mathbb{A})\}$. It is compact in $Z(\mathbb{A})\setminus GL(2,\mathbb{A})$. But is it compact in $GL(2,\mathbb{A})$? I don't know how to show it. How can i show if it is compact or not? If it is compact, then could you give a counter example for the statement i gave at the beginning of my question?