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. 



evaluation map $ev_t$ on loop space
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 $\mathbb R^n$. 


