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#58;#92;mathbf#123;Open#125;(X)#92;mbox#123;op#125;#92;rightarrow#92;mathbf#123;Set#125;. 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#92;subseteqU a function F(U)#92;rightarrowF(V), called restriction from U to V and denoted by s#92;rightarrows#124;#95;V. I will write Psh(X) for the category of presheaves on X. Examples 3.1 i. Let F(U) = {continuous functions U#92;rightarrow#92;mathbb#123;R#125;}; restriction is restriction.
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