Constructible subsets of a scheme

Let $X$ be a topological space, for instance the underlying space of a scheme (Quasi-compact and quasi-separated schemes). An open subset $U\subseteq X$ is retrocompact if for every quasi-compact open subset $V\subseteq X$ the intersection $U\cap V$ is quasi-compact as a topological space. A subset $Z\subseteq X$ is constructible if it is a finite union of subsets of the form $U\cap (X\setminus V)$ with $U,V\subseteq X$ retrocompact open. By convention the empty union is allowed, so $\mathrm{\varnothing }$ is constructible, and every retrocompact open is constructible, by taking $V=\mathrm{\varnothing }$.

A subset of a scheme is called constructible when it is constructible in the underlying topological space. Since a morphism of schemes is continuous, the preimage of a constructible subset is constructible whenever the preimages of retrocompact opens are retrocompact; this is checked where it is used and is not part of the definition.

For an affine scheme $X=\mathrm{Spec}A$ (The underlying space of an affine spectrum) the notation $D(f)$ for a basic open and $V(g_1,\dots ,g_m)$ for a vanishing set is available, and there the retrocompact opens are exactly the quasi-compact opens: the space $\mathrm{Spec}A$ is quasi-compact, so a retrocompact open is quasi-compact; conversely a quasi-compact open is a finite union of basic opens, and basic opens are spectra of rings and hence quasi-compact, so intersections of quasi-compact opens are finite unions of basic opens and quasi-compact. A subset of $\mathrm{Spec}A$ is therefore constructible exactly when it is a finite union of sets $D(f)\cap V(g_1,\dots ,g_m)$; the description by finitely many $D(f)\cap V(g_1,\dots ,g_m)$ is the one used in the constructibility results on this page.