What about $GL(n,\mathbb C)$? Is it open, dense in $M(n,\mathbb C)$? What about $GL(n,\mathbb C)$? Is it open, dense in $M(n,\mathbb C)$?

Yes, it is both open and dense.  One way to show it is open is by considering the determinant.  Alternatively, the set of invertible elements of any Banach algebra is open (and $M(n,\mathbb{C})$ is a Banach algebra if you give it any submultiplicative norm), which follows from the fact that if $a$ is invertible and $\|a-b\|\lt \frac{1}{\|a^{-1}\|}$, then $b$ is invertible.  One way to show it is dense is to consider the fact that $a\in M(n,\mathbb{C})$ has only a finite set of eigenvalues.
