Let X be a topological space. (Following tradition, I will switch from my previous convention of using X to denote an object of a topos.) Write Open(X) for its poset of open subsets. A presheaf on X is a functor F: mathbf Open(X)op→ mathbf Set. It assigns to each open subset U a set F(U), whose elements are called sections over U (for reasons to be explained). It also assigns to each open V⊆U a function F(U)→F(V), called restriction from U to V and denoted by s→s|_V. I will write Psh(X) for the category of presheaves on X.

Examples 3.1 i. Let F(U) = {continuous functions U→ℝ}; restriction is restriction.