Let $E$ be locally convex topological vector space. Let $c^\infty E$ denote the same vector space equipped with the $c^\infty$-topology (i.e. the finest topology on it, s.t. all smooth curves $\mathbb{R} \rightarrow E$ are continous, see "The convenient setting of global analysis" by A.Kriegl and P.Michor, I.2.12). If now $E,F$ are two such spaces, it may happen that the identity $c^\infty(E \times F) \rightarrow c^\infty(E) \times c^\infty(F)$ is continous but not a homeomorphism (see I.4.16 of the same reference).

I am wondering whether it is a homeomorphism in the following situation: Let $M$ be a smooth, finite-dimensional (but not necessarily compact) manifold and $V = V_1 \oplus V_2$ the Whitney sum of two finite-rank vector bundles over $M$. Then we have $\Gamma_c(V) \cong \Gamma_c(V_1) \times \Gamma_c(V_2)$ as vector spaces and I think this is also an isomorphism of locally convex spaces if all these spaces are equipped with the usual strict inductive limit topology. But are the spaces $c^\infty \Gamma_c(V_1) \times c^\infty\Gamma_c(V_2)$ and $c^\infty\Gamma_c(V)$ homeomorphic ? As far as I can see, they are linearly diffeomorphic (since the isomorphism maps smooth curves to smooth curves) but I don't know, under which (additional) conditions this map is also a homeomorphism.