You can't cover an arbitrary locally ringed space with open affine schemes? Should I rewrite my question?

Are you saying that there is a bijection between sheaf maps $\alpha : \mathscr{F} \to \mathscr{G}$ and continuious maps $f : {\rm Spe}(\mathscr{F}) \to {\rm Spe}(\mathscr{G})$?

@Mariano: I fixed the title.

Atleast on $\mathbb{A}^n$, I like to think of the open sets as the sets on which regular functions do not vanish. The fact that there are not enough regular functions to make the Zariski topology hausdorff just says to me that algebraic varieties are more like holomorphic manifolds than differential manifolds

@Robin Chapman: I think it is proved in the last chapter of Do Carmo's book "The differential Geometry of curves and surfaces"