First of all sorry for the (possible) incorrect english. I don't know english very well.

I'm with a doubt about topology of maps between fibres of vector bundles.

Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all linear maps $T:E_x \rightarrow F_y$, where $E_x$ is the fiber over $x$ and $F_y$ is the fiber over $y$.

I want to know a way to define the topology of this set!

I need this topology for this question: Consider $f: E \rightarrow F$, a map that preserves each fiber and its restriction to each fibre, $f_x : E_x \rightarrow F_y$, is differentiable. The differential of $f_x$ calculated in a vector $v \in E_x$ is the linear maps $df_x(v):E_x \rightarrow F_y$. I want to say that $f$ is a $C^1$ map if the function $v \in E \rightarrow df_{\pi(v)} (v)$ is continuous. And for this I need a topology for the set defined above.

Anybody know how to define the topology? What means two of these maps are close to each other

Note that $df_x (v)$ is not necessarily a homomorphism, because this is only defined over the fiber that contains $v$.

Thanks