This is the first exercise in zeroth chapter of Lie groups beyond introduction by A. knapp: show that exp maps $\mathfrak{gl}(n, \mathbb{C})$ onto $GL(n,\mathbb{C})$ but does not map $\mathfrak{gl}(n, \mathbb{R})$ onto $GL(n,\mathbb{R})$.
For the second part, I noted that $exp(\mathfrak{g})$ generates the connected component of the identity, and since $GL(m,\mathbb{R})$ is not connected under the subspace topology induced from $\mathbb{R}^{n^2}$, the desired conclusion follows.
Is there a sleek proof for the first part, other than using Jordan Canonical form. Moreover, what is the topology on $GL(n,\mathbb{C})$? Is it connected when considered inside $\mathbb{R}^{2n^2}$? What is the topology one usually means when they refer to $GL(n,\mathbb{C})$ ? is it considered inside $\mathbb{R}^{2n^2}$ or $\mathbb{C}^{n^2}$?