Q:

Morphism of schemes $f\colon X\to Y$ associated to a continuous map of the underlying spaces $|X|\to |Y|$

I am sorry for asking two questions in one but they are strongly related. 

What is an example of (affine?) schemes $X=(|X|,\mathcal{O}_X)$ and $Y=(|Y|,\mathcal{O}_Y)$ and a map of topological spaces $|f|\colon|X|\to |Y|$ that cannot be promoted into a map $f\colon X\to Y$ of schemes?

I guess something like $exp:\mathbb{R}\to\mathbb{R}$ is an example but I cannot prove that it is an example.

What is an example of (affine?) schemes $X=(|X|,\mathcal{O}_X)$ and $Y=(|Y|,\mathcal{O}_Y)$ and a map of topological spaces $|f|\colon|X|\to |Y|$ that can be promoted into a map $f_1\colon X\to Y$ of schemes and into a map $f_2\colon X\to Y$ a map of schemes with $f_1\neq f_2$?

A:

Just take $X=\mathrm{Spec}(K)$ and $Y=\mathrm{Spec}(L)$ for two fields $K,L$.
There is a unique map $|X| \to |Y|$. The morphisms $X \to Y$ correspond to field homomorphisms $L \to K$. There may be no such homomorphisms, but there may be also many of them. (For example, consider $\mathbb{Q}(\sqrt{2}) \to \mathbb{Q}$ or $\mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2})$).

A:

1) Take $X=Y=\mathbb A^1_\mathbb C$, so that $|X|=\mathbb C\sqcup \{\eta\}$.
 Any permutation of $|X|$ fixing the generic point $\eta$ induces a homeomorphism $f:|X|\to |X|$ .
If the induced permutation on the subset of closed points  $\mathbb C\subset |X|$ is not continuous in the classical topology, then the map $f$ cannot come from a scheme morphism $X\to X$.
As an example you can take the permutation exchanging $0$ and $1$ and fixing everything else.  
2) Given a field $k$ and a non-trivial automorphism $\phi:k\to k$, the induced scheme morphisms $\phi^*, Id^*:\text {Spec}(k)\to \text {Spec}(k)$ are different but induce the same homeomorphism (of one point spaces!) $|\text {Spec}(k)|\to |\text {Spec}(k)|$ 

