More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles.
Now consider the subspace consisting of $C^\infty(M)$-linear maps $B:\Gamma(E_1) \to \Gamma(E_2)$ satisfying $\ker(A) \subseteq \ker(B)$.
Is the subset of maps $B$ as above satisfying $\ker(A) = \ker(B)$ open (in some $C^p$ topology) inside the set of maps such that $\ker(A) \subseteq \ker(B)$?
I believe this should be true as reducing modulo $\ker(A)$, the question translates into a map nearby an injective map (the map induced by $A$) should still be injective, but I am not entirely sure if this idea still holds in the infinite-dimensional setting.
Thanks in advance!
