We want to show that for a given $n\in\mathbb{N}$, the set of all the $\infty\times n$ matrices $Mat_{\infty,n}(\mathbb{C})$ of max rank $n$ form a contractible space.
This set is obtained by taking the limit on $k \geq n$ in $Mat_{k,n}(\mathbb{C})$ with max rank $n$, being all $Mat_{j,n}(\mathbb{C}) \subset Mat_{j+1,n}(\mathbb{C})$ by adding a new raw of zeros. Our space is endowed with the Zarisky topology, given by induction on the Zarisky topology of the $Mat_{k,n}(\mathbb{C})$'s .
I don't think this is a CW-complex so I don't know which strategies to use. If we remove the $\infty$ the result is no more true, really similar to $S^n$ and $S^{\infty}$.
Thanks!