how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

Question

could any one tell me which of the following is/are compact subset?

1. $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$

2. $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$

3. $S=\{A\in M_n(\mathbb{C}): AA^*=A^*A=I\}$

I pointed out that only $2$ is compact by the map $Ap:\mapsto A-A^*$ gives $S=p^{-1}(O)$ hence closed and clearly this set is bounded.

$3$ is just $O_n(\mathbb{C})$ I can prove that it is not bounded so not compact