TITLE: Constructible topology on schemes
QUESTION [9 upvotes]: 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?

REPLY [6 votes]: 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.