Constructible topology on schemes In EGA IV Grothendieck introduced notion of constructible topology. Is it only interesting gadget or can it be use for some practical purposes in algebraic geometry?
 A: Julien Sebag tells me that the constructible topology is useful for the study of the Grothendieck ring of varieties. More precisely, it is relevant to the following question: "if $k$ is a field and $X$ is a $k$-variety with a birational endomorphism $X--\to X$inducing an isomorphism between open subsets $U$ and $V$, are $X\setminus U$ and $X\setminus V$ piecewise isomorphic? You may read about this in Liu and Sebag - The Grothendieck ring of varieties and piecewise isomorphisms.
Another place where the constructible topology is essential is in motivic integration, where constructible sets play the role of the measurable sets of usual integration theory.
A: In Hochster - Prime Ideal Structure in Commutative Rings (MSN), the author uses it to characterize spectral spaces. This in turn is used in Huber's work on Adic spaces, cf. Huber - Étale cohomology of Rigid Analytic Varieties and Adic Spaces (MSN) and Scholze - Perfectoid spaces (MSN).
A: The constructible topology is used in Bhatt and Scholze - The pro-étale topology for schemes, henceforth [BS], in an essential way (as far as I understand it).
The pro-étale topology is a generalization of the étale topology where $\overline{\mathbb{Q}}_\ell$-étale cohomology is actually the sheaf cohomology (in the naive sense) of the constant sheaf $\overline{\mathbb{Q}}_l$ (rather than $\varprojlim_n H^i(X_{\text{et}}, \mathbb{Z} / \ell^n \mathbb{Z}) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$). The point is to enlarge the étale site to include "limits" of coverings, (so for example $\mathrm{Spec}(k^{\text{sep}})$ is always in the pro-étale site of $Spec(k)$, and $\mathrm{Spec}(k^{\text{sep}}) \to \mathrm{Spec}(k)$ is a pro-étale covering).
The Zariski version of this is a little more accessible. In this pro-setup, there is an "initial" "Zariski" covering $X^Z \to X$, which is the "intersection" of all Zariski coverings. The closed points $(X^Z)^c$ of $X^Z$ are in bijection with the points of $X$, and $(X^Z)^c$ equipped with the subspace topology (via the canonical inclusion $(X^Z)^c \subseteq X^Z$) is homeomorphic to $X$ equipped with the constructible topology [BS, Lemma 2.1.10].
The fact that the constructible topology is quasi-compact is used to produce a "contractible" covering of $(X^Z)^c$ in the site of pro-finite topological spaces [BS, Def.2.4.4, Thm.2.4.5, Exa.2.4.6, Exa.4.1.10], and this is one of the main steps in the construction of a "contractible" pro-étale covering of an arbitrary scheme [BS, Lemma 2.4.9, Prop.4.2.8].
The existence of "contractible" coverings is reason the pro-étale topology works [BS, Def.3.2.1, Prop.3.2.3, Def.3.1.1, Lem.3.3.2, Prop.3.3.3, Prop.5.5.4, Prop.5.6.2]. The fact that the constructible topology is quasi-compact is one of the main ingredients for this existence of "contractible" coverings.
