Topological reconstruction theorems over uncountable algebraically closed fields

Working over uncountable algebraically closed fields, we extend the theorems of Kollár-Lieblich-Olsson-Sawin on reconstructing varieties from their Zariski topological spaces. In particular, we adapt their results to arbitrary quasi-projective varieties in arbitrary characteristic, and thus we give positive answers to each of the relevant speculations' made by the original authors in our setting. Our proofs use techniques from model theory: in particular, we employ a general model-theoretic setting for algebro-geometric reconstruction problems, known as the Zilber trichotomy for ACF-relics’.