Prime Spectrum Topology Let $X$ be a scheme, I want to show that there exists a unique scheme map $f:X\to \text{spec}(\mathbb{Z})$. Thus, far I have established that such a map, on the topology, is unique. Let $V$ be an affine open subset of $X$. Is it true that $V = f^{-1}(W)$ for some open set $W$ in $\mathbb{Z}$? Follow up question, if not true, does there exist a base for the topology of $X$ consisting of affine open sets which have this form? (Do not use anything other than the definitions, I am trying to prove a theorem from the ground up.) 
 A: Nope.
For any $W\subset \operatorname{Spec}(\mathbb{Z})$, $f^{-1}(W)$ is a union of fibers. Of course not every affine open of $X$ will be a union of fibers. In particular, if $V$ is a union of fibers, you can usually consider a smaller piece which is not a union of fibers.
I would first suggest you play around with some simple examples. Start with say $X = \operatorname{Spec}(R[x])$ for some ring $R$ (first maybe let $R$ be a field, then more general rings...?). Can you come up with a morphism $\operatorname{Spec}(R[x])\rightarrow \operatorname{Spec}(\mathbb{Z})$? Can you come up with any others? Then perhaps consider some nonaffine scheme $X$. What's the simplest non-affine scheme people generally first learn about? Maybe projective space? Take the projective line, and try to come up with a morphism $\mathbb{P}^1_R\rightarrow \operatorname{Spec}(\mathbb{Z})$...etc. How does one even define a morphism from a non-affine scheme to an affine scheme? (Hint: First define it on affine pieces of the domain, then show they agree on intersections, and "glue").
After that, I would suggest you attempt to prove your thing as follows:
First consider the case where $X$ is affine, say $X = \operatorname{Spec}(R)$. Show that there exists a unique morphism $\operatorname{Spec}(R)\rightarrow \operatorname{Spec}(\mathbb{Z})$. In general, write $X$ as a union of affine schemes $\operatorname{Spec}(R_i)$, and show that the unique morphisms $f_i : \operatorname{Spec}(R_i)\rightarrow \operatorname{Spec}(\mathbb{Z})$ agree on intersections, and thus glue to give a unique morphism $X\rightarrow \operatorname{Spec}(\mathbb{Z})$
A: If $X=\mathbb A^1_\mathbb Z=\operatorname{Spec }\mathbb Z[T]$, the open affine subscheme $V=D(T)(=X\setminus V(T))$ is an affine open subset $U\subset X$ containing no fibre of the unique scheme morphism $\mathbb A^1_\mathbb Z \to\operatorname{Spec }\mathbb Z$.
Thus $V$ is  certainly not of the form $f^{-1}(W)$ for any $W\subset \operatorname{Spec }\mathbb Z$ .
