Using analytic continuation, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined . ( This does not imply that all the restriction maps of this sheaf are injective !)
12.
A presheaf on " X " chooses a set for each of the four open sets of " X " and a restriction map for each of the nine inclusions ( five non-trivial inclusions and four trivial ones ).
13.
In molecular biology, restriction maps are used as a reference to engineer plasmids or other relatively short pieces of DNA, and sometimes for longer genomic DNA . There are other ways of mapping features on DNA for longer length DNA molecules, such as mapping by transduction.
14.
The "'constant presheaf "'with value "'Z "', which we will denote " F ", is the presheaf which chooses all four sets to be "'Z "', the integers, and all restriction maps to be the identity . " F " is a functor, hence a presheaf, because it is constant . " F " satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set.
