Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$ using $\mathcal{C}^1$ Whitney Topology.
Now, consider $S\subset M$ a compact submanifold of $M$ with boundary such that $\text{dim}S=\text{dim}M$, using the same process we can put a topology in $\mathcal C^\infty(S,N)$ using the $\mathcal{C}^1$ Whitney Topology. There is a natural continous projection of $\mathcal C^\infty(M, N)$ on $\mathcal C^\infty(S,N)$, definided by
\begin{align*} \pi: \mathcal C^\infty(M, N) &\to \mathcal C^\infty(S,N)\\ f&\mapsto \left.f\right|_{S}. \end{align*}
My Question: Is $\pi$ an open map or at least a quocient map?