This is Exercise I-5 in The Geometry of Schemes by Eisenbud and Harris.
Let $X$ be the two-element set $\{0,1\}$, and make $X$ into a topological space by taking each of the four subsets to be open. A sheaf on $X$ is thus a collection of four sets with certain maps between them; describe the relations among those objects. ($X$ is actually $\mathrm{Spec}\,R$ for some rings $R$; can you find one?)
I am not exactly sure how a solution to this exercise should look like. I am new to sheaves, and maybe the definition needs some clarification.
So the topology is $\tau=\{\emptyset, \{0\}, \{1\}, X\}$.
Now I can write down all the restriction maps, and associated sets $\mathcal{F}(U)$ for $U\in\tau$.
What exactly is $\mathcal{F}(\emptyset)$ for example, or $\mathcal{F}(X)$? Are those just sets, and it does not matter, or is there a way to figure out more information about these sets?
The same goes for the restriction maps $\operatorname{res}_{\,U,V}$
What exactly are the sections of $\mathcal{F}(U)$? These seem to be "functions", but the definition of a presheaf in the book does not really specify this. So what is the domain, or codomain?
By the sheaf axiom we would have that for example the sections $f_X\in\mathcal{F}(X)$ and $f_0\in\mathcal{F}(\{0\})$ have to be equal on the intersection $X\cap\{0\}=\{0\}$.
Are sections in $\mathcal{F}(X)$ functions $f:X\to X$ and sections in $\mathcal{F}(\{0\})$ functions $f:\{0\}\to X$, or does this simply not matter?
As far as the additional question goes, I have noticed that this was asked before: Exercise I-5 of Eisenbud-Harris
But this is not the point of my question here. I want to get more familiarity with the definition of a sheaf.
Thanks in advance for any clarification/insights on sheaves.