Sheaves and affine varieties

Definition 3. A presheaf F is a sheaf if for every collection f/ of open sets in X with U = UUi, the diagram 

When we pull this high-flown terminology down to earth, it says this. 

1) If xi,X2 G F(U) and for all i, resu,Uixi = resu,Uix2, then xi = X2. (That is, elements are uniquely determined by local data.) 

2) If we have a collection of elements Xi G F U ) such that resui Uir Ujxj — vesu UiDUjXj for all and j then there is an x G F(U) such that resu,Uix = xi for all i. (That is, if we have local data which are compatible, they actually patch together to form something in F(U).) 

Example G. Let X and Y be topological spaces. For all open sets U C X, let F(U) be the set of continuous maps U — Y. This is a presheaf with the restriction maps given by simply restricting maps to smaller sets it is a sheaf because a function is continuous on UUi if and only if its restrictions to each Ui are continuous. 

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