evaluation map $ev_t$ on loop space Considering parameter of $S^1$ as $t$, we define.
$$ev_t: C^\infty(S^1, \mathbb R^n)\to \mathbb R^n$$
$$ev_t(\gamma):=\gamma(t)$$
I am looking for a possible topology on $C^\infty(S^1,\mathbb R^n)$ which makes $ev_t$, an open map. 
 A: The (compact) $C^\infty$-topology makes $ev_t$ an open map. Namely, any open neighborhood $U$ of $f$ is described by: You may deviate from $f$ uniformly by a positive constant for a chosen finite set of derivatives and still stay in $U$; no conditions on the derivatives outside this set.
So for the 0-th derivative you may always deviate a positive amount from $f(t)$.
Thus $f(t)=ev_t(f)$ is an inner point of $ev_t(U)$ in $\mathbb R^n$. 
A: I suspect it will suffice to find a topology for whicg $ev_1$ is an open map. (In any reasonable topology, loop reparametrization should define a self-homeomorphism of $C^{\infty}(S^1,\mathbb{R}^n)$.)
For a Lie group $G$, $$LG:=C^{\infty}(S^1,G)$$ is its loop group. We find that $$ev_1:LG\rightarrow G$$ is a surjective group morphism with kernel the based loop group $$\Omega G:=\{\gamma\in LG:\gamma(1)=e\}.$$ So, I think the trick is to find a topology on $LG$ for which $$LG\rightarrow G=LG/\Omega G$$ is a principal $\Omega G$-bundle. I believe this will render $$ev_1:LG\rightarrow G$$ an open map.
For compact connected semisimple $G$, a standard reference on $LG$ and $\Omega G$ is Pressley-Segal. In this case, $ev_1$ should be a principal $\Omega G$-bundle. Some of the relevant constructions may also work for $G=\mathbb{R}^n$.
