Presheaves that are not sheaves
This is a post about pre-sheaves that are not sheaves. The two properties that a sheaf satisfies that a presheaf does not, are the “Gluability” axiom and the identity axiom.
Gluability- Over the open set , consider the set , which is the set of all bounded continuous functions. Sections which have the same restriction to intersections of open sets sometimes cannot be glued together to give global sections; i.e. they cannot be glued together to give a section defined on the whole topological space that is bounded. For instance, consider the function . Consider the cover of . The restriction of on each such bounded interval is bounded. Hence, . However, on gluing together over the whole real line, we get an unbounded function.
Another example seems slightly trickier to me, and requires knowledge of complex analysis. Vakil’s book says that “holomorphic functions that admit a square root” form a presheaf and not a sheaf. Note that we just need to consider functions that admit a square root, and not the square roots themselves. Take the function . It admits a square root on both and . Both and are open sets. Moreover, the restrictions of the function on the overlap of the two sets agree at each point. However, when we glue the two parts together, we get defined on the whole complex plane, which is holomorphic, but does not admit a square root.