Is the forgetful functor from locally ringed spaces to topological spaces a full functor? Faithful? What about when restricted to schemes?

Question

In detail, given locally ringed spaces $X,Y$ with underlying topological spaces $X_0,Y_0$,

• can every continuous map $f_0 : X_0 \rightarrow Y_0$ lift to a morphism of ringed spaces $f : X\rightarrow Y$?
• if such a lift exists, is it unique?

My suspicion is that the answer to the above is no. Then, my 2nd question is, is the above true when restricted to schemes $X,Y$?

Answer 1

This is false. For any scheme $X$, there is only one morphism $X \rightarrow Spec(\mathbb{Z})$. However, there can be multiple different continuous functions $X \rightarrow Spec(\mathbb{Z})$.

Even if a lift exists, it might not be unique. For example, consider $Spec(\mathbb{C}) \rightarrow Spec(\mathbb{C})$. There is only one continuous function, but it can be lifted in two ways: identity and complex conjugation.