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 $f(Y) = T$?

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.
