I'm not a mathematician (and I'm not a native speaker either) so I apologize if some of the things I'm saying are inaccurate or incorrect.
I'll try to be as intuitive as possible. Suppose I have a sheaf of parallel straight lines. If I understood it correctly, the union of those generates a surface. However, there is a difference between the sheaf of lines themselves and the surface, right? For example, take a straight line that is conplanar to the ones of the sheaf mentioned above but has a different slope. This line would not belong to the sheaf but still "belong" to the surface generated by the union of the lines of the sheaf, am I right? I'd just like to know if what I've said so far makes sense.
Additionally, say I have a non-convex and continuous function F and I want to define the set of all the functions G obtained as the horizontal shifting of F in a given interval [a, b]. How can I do that? Does what I've said about the sheaf of straight lines also hold for the set of functions mentioned here (i.e. there is a difference between the set and the surface I can generate from the "union of the functions")? Thank you very much!