I have problem showing that this sets are path connected. It's a little abstract for me to work in $Gl_n$.
I have to prove that the following sets are path connected in $C^{n^2}$. a) diagonalizable complex $nxn$ and invertibles matrices.
b) hermitian matrices
c) unitary matrices ( Hint : it's similar to a diagonal matrix)
d) Using the fact that every complex matrix $nxn$ it's the propduct of an hermitian matrix positive definite and an unitary matrix, prove that $ Gl_n (C)$ it's path connected in $C^{n^2}$
My solution I realized that in some cases (maybe all) this sets are open, so it's enough to prove connectedness. And for this it's enough to show a continuous surjection from a connected set (like a subset of R^n) for example. In a) the surjection is given by $(a_1,...a_n) $ to the matrix with that diagonal , with $a_i \ne 0$ it's continuous but not connected, but I think that I can fix the problem defining a lot of maps (restricting the domain in several ways such that all the new domains are connected), such that the image of all of them always intersect, and then the union will be the set of matrices .
But the others? I have no idea what can I do!!