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I don't understand why the claim in your second paragraph is an axiom. Doesn't it follow trivially from your definition?

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The "sheaf axioms" define what it means for a "rule" associating "data" to open subsets to be a sheaf, and I was just trying to illustrate them with the example of "F". (Perhaps calling them "properties" or "laws" -- or even an interface or typeclass! -- might help?)

In general, proving that something that looks like a sheaf really is one may be nontrivial. :)

In the special case that I outlined above, it certainly is easy to show that F satisfies those axioms, as you point out. And it is a sheaf (the sheaf of continuous real-valued functions on R) precisely because it does.

Sorry, I meant the claim about f and g. Assuming you meant that F(I) should be continuous functions, you can construct an h from f,g to be cont on I and J, no? So it's not an axiom. Just making sure I understand correctly...

Again, that was just an example. F is just one possible sheaf on the real line, and in the case of F, yes, continuous functions can be stitched together.

You could define A(I) = { } for a trivial example of a different sheaf A where the "data" (always an empty set, regardless of I) is very different from what it was in the case of F (the set of continuous functions on I).

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