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.
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.