Is there an easy way to prove that every subgroup of $SU(2)$ is closed? If there isn't, is there a reference for this fact?
I thought that this was left as an exercise on the book on compact Lie groups by Sepanski, but I cannot find the reference. Thanks.
No. Take, for instance, the group of those matrices of the form$$\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix},$$with $\theta\in\mathbb Q$.