I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper "Contractibility of the space of rational maps". The paper for the most part is beyond me, but there seems to be a theorem that if $Y$ is a connected affine scheme covered by opens of $\mathbb{A}^n$ for some $n$. The space $Map_{rat}(X,Y)$ is contractible. One question that I had and I wasn't able to find an explicit description in the paper. What is the topology on $Map_{rat}(X,Y)$? Is it the same topology induced by the compact open topology on the space of continuous maps?

On paragraph above section 0.5.7., it mentions that this might be true under milder condition that requires $Y$ to be only smooth and birational.

I was wondering whether there are further results in this direction. I was particularly interested in the space of rational maps from $\mathbb{A}^m$ to something like $Sym^{\infty}(\mathbb{P}^r)$, is there any way to figure out its homotopy type maybe by considering a specific CW complex structure on $Sym^{\infty}(\mathbb{P}^r)$ and then seeing how each part is glued to each other after taking the $Map_{rat}(\mathbb{A}^m,-)$.