# Difference between sheaf of parallel straight lines and the surface generated from the union of its lines

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!

## migrated from mathematica.stackexchange.comMar 23 '18 at 16:45

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• Could you please define what you mean by a "sheaf of lines"? The mathematical meaning of "sheaf" that I am most familiar with does appear to be what you are talking about. – Paul Sinclair Mar 24 '18 at 2:16
• Oh I'm sorry. The set of all the straight lines with a given slope. An example would be y=x+k – Sith Mar 24 '18 at 14:22
• The union of all straight lines with a given slope is all of space (whichever dimension of space you are working in). It is not a surface. – Paul Sinclair Mar 24 '18 at 14:29
• Okay, still, the set of those straight lines and the space would not be the same unless do the union, right? – Sith Mar 24 '18 at 16:29
• Techniqually, no. The set of lines is a set of sets of points, while space is a set of points. However, in geometry, one tends to ignore such fine points of set theory. – Paul Sinclair Mar 25 '18 at 4:48