$GL(n,\mathbb{R})$ can be seen as $\mathbb{R}^{n^2}$ with points of non invertible matrix removed which form "holes" in $\mathbb{R}^{n^2}$. These holes seems to be dense in $\mathbb{R}^{n^2}$ which will mean any neighbourhood of $GL(n,\mathbb{R})$ will contain holes, even for the neighbourhood that is homeomorphic to some neighbourhood of the oringin of $\mathbb{R}^{n^2}$. It is strange that a region with holes is homeomorphic to a region without holes.
Is this an example of a region with holes is homeomorphic to a region without holes, or is non invertible matrix not dense in $\mathbb{R}^{n^2}$?
$GL_{n}(\mathbb{R})$ is open in $\mathbb{R}^{n^2}$. This is a consequence of the determinant map being continuous. Also tells you that the set of non invertible matrices cannot be dense. The invertible matrices on the other hand are.