If rank($A$)$>k$ where $A$ is a $n \times n$ matrix then is the set of such matrices open? ($k<n$)
If $det(A) \ne 0$ then we know that rank($A$)=$n$ then collection of such matrices is open in $M_n(\mathbb{R})$.
Can we think of a continuous map such that $M_n(\mathbb{R}) \to \mathbb{R}$ such that it maps the rank of the matrix? Or something similar to it?
