Intersection of quasi-compact open subsets of an affine scheme Let $X = \mathrm{Spec}(A)$ be an affine scheme.
Let $U$ be a quasi-compact open subset of $X$.
Then there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = U$, right?
We are interested in a similar result(if any) on an arbitrary intersection of quasi-compact open subsets of $X$.
Let $(U_i)_{i\in I}$ be a family of quasi-compact open subsets of $X$.
Let $T = \bigcap_{i\in I} U_i$.
Do there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = T$?
 A: I don't know what is the motivation of the question. But the answer is yes. 
In an affine (hence separated) scheme $X$, any affine open subset is retro-compact hence constructible. So any quasi-compact open subset of $X$ is constructible in $X$. Any intersection of constructible subsets of $X$ is pro-constructible (EGA IV.1.9.4), and, as $X$ is quasi-compact and quasi-separated, any pro-constructible subset of $X$ is the image of an affine scheme (EGA, IV.1.9.5(ix)). 
Edit Sketch of the proof of EGA, IV.1.9.5(ix): 


*

*First a constructible subset of $X$ is the image of an affine scheme $X'$ (EGA, IV.1.8.3). It is enough to prove it for a locally closed subset $U\cap (X\setminus V)$, so a quasi-compact open subset of an affine scheme $X\setminus V$, write it as a union of affine open subschemes $U_1, U_2...$ and take $X'$ the disjoint union of the $U_i$'s). 

*(EGA IV.1.9.3.2) Now for a pro-constructible subset $\cap_i C_i$, if $C_i$ is the image of $f_i: X'_i=\mathrm{Spec}(A_i)\to C_i$, define $A'$ as the direct limit of tensor products of finitely many $A_i$'s and consider the canonical morphism $f: \mathrm{Spec}(A')\to X$ (each $A_i'$ is an $A$-algebra via $f_i$), show $f(X')=\cap_i C_i$. This part is harder.  
A: No. Delete countably many points from the affine line over an uncountable field. This set is not constructible and cannot be the image of an affine.
