I'm looking at the proof of Proposition 2.4 in Atiyah-Bott Yang-Mills on Riemann surfaces:
Proposition: Let $P$ be a principal $G$-bundle over a compact Riemann surface $M$ with gauge group $\mathscr{G}(P)$, and let $BG$ be the classifying space for $G$. Then (up to homotopy) $$B\mathscr{G}(P) = \text{Map}_P (M,BG)$$ where $\text{Map}_P (M,BG)$ is the space of maps inducing $P$ on $M$.
The proof proceeds by showing $\text{Map}_G(P,EG)$, the space of $G$-equivariant maps from $P$ to $EG$, is a principal $\mathscr{G}(P)$-bundle over $\text{Map}_P(M,BG)$. The claim is that since $\text{Map}_G(P,EG)$ is contractible, $\text{Map}_P(M,BG)$ is a model for the classifying space of $\mathscr{G}(P)$.
My questions are:
- Are the topologies on these spaces of maps the induced topologies coming from the compact-open topology?
- Why is $\text{Map}_G(P,EG)$ contractible (presumably with respect to the compact-open topology)?