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 $M(n,\mathbb C)$?

The amount of algebraic geometry needed to prove the denseness of $GL_n({\Bbb C})$ is indeed very minimal.
The topology of $M_n({\Bbb C})$ comes from its obvious identification with ${\Bbb C}^{n^2}$. The complement is the zero set of the determinant which is a polynomial in the $n^2$ entries of the matrix. The zero set of a polynomial cannot have inner points: if a polynomial vanishes on an open ball, its Taylor expansion centered at any point of the ball is identically 0, and by analicity the polynomial vanishes identically. Therefore there are no proper closed subsets of $M_n({\Bbb C})$ containing $GL_n({\Bbb C})$.
