Let ${\cal M}_k $ be the set of linear maps that select $k \times k$ minors, and define $\delta_k(A) = \max_{L \in {\cal M}_k} | \det L(A) |$, note that $\delta_k$ is continuous and $\operatorname{rk} A = k$ **iff** $\delta_k(A) >0 $ and $\delta_{k+1}(A) = 0$ (define the latter to be zero if $k=\min(m,n)$ for convenience).
Note that a matrix has rank $\le k$ **iff** $\delta_{k+1}(A) = 0$ and so the set of such matrices is closed.
Using the SVD, you can show that any matrix of rank $\le k$ can be approximated by matrices of rank $k$ and so $\overline{R_k} = R_1 \cup...\cup R_k$.
