Let $E$ be a Banach space, $X$ a locally-compact metric space, equip $C(X,E)$ with the compact-open topology. Let $F:E \rightarrow E$ and consider the induced map $F_{\star}(f):=F \circ f$ on $C(X,E)$. When is $F_{\star}$ open?
Update More generally, what is an example of an open mapping $G:C(X,E)\to C(X,E)$ which is not a homeomorphism (possibly not continuous)?
Update: I think this should work, but, I feel that something is wrong with the argument...
Let $F:E\rightarrow E$ be open and admit a continuous-left inverse. For every compact $K\subseteq X$ and over open $U\subseteq E$ (both non-empty) we know that $\{f\in C(X,E):f(K)\subseteq U\}$ is a basic open subset of $C(X,E)$ for this topology. Then $$ F_{\star}(\{f\in C(X,E):f(K)\subseteq U\})= \{g \in C(X,E): (\exists f \in C(X,E))\, g=F\circ f \mbox{ and } g(K)\subseteq F(U)\} = \{g \in C(X,E): g(K)\subseteq F(U)\} , $$ but since $F$ is open then $F(U)$ is open. Hence, $F_{\star}(\{f\in C(X,E):f(K)\subseteq U\})$ is a basic open subset of $C(X,E)$ for this topology.