In the text of Algebraic geometry,Hartshorne defines presheaf like this.
And as the above text says,this can be rephrased by using category word.
i.e. presheaf is contravariant functor from the category $Top(X)$ to the category $Ab$
($Top(X)$ is category whose objects are open subsets of X, and only morphisms are the inclusion maps,and $Ab$ is category of abelian groups,see image for details)
I have trouble in this definision.
question
If $F$ is contravariant functor from the category $Top(X)$ to the category $Ab$ ,how to show $F(∅)=0$.
This may be a trivial problem, but I have no idea so any help would be appreciated.