Reconstructing a variety from its topology (after Koll\'{a}r, building on earlier work of Lieblich and Olsson)

The underlying Zariski topological space |X| of an algebraic variety or, more generally, a scheme X tends to have few open subsets in comparison to topologies that underlie structures appearing in differential geometry or geometric topology. Thus, intuitively, |X| is a weak invariant of X, and this intuition is confirmed by low-dimensional examples: for an algebraic curve C, the proper closed subsets of |C| are the finite subsets of closed points, so |C| does not see much beyond the cardinality of the algebraic closure of the base field. A more surprising example was constructed by Wiegand and Krauter [WK81, Cor. 1]: for primes p and p, there is a homeomorphism