  We study Torelli-type theorems in the Zariski topology for varieties of
dimension at least 2, over arbitrary fields. In place of the Hodge structure,
we use the linear equivalence relation on Weil divisors. Using this setup, we
prove a universal Torelli theorem in the sense of Bogomolov and Tschinkel. The
proofs rely heavily on new variants of the classical Fundamental Theorem of
Projective Geometry of Veblen and Young.
  For proper normal varieties over uncountable algebraically closed fields of
characteristic 0, we show that the Zariski topological space can be used to
recover the linear equivalence relation on divisors. As a consequence, we show
that the underlying scheme of any such variety is uniquely determined by its
Zariski topological space. We use this to prove a topological version of
Gabriel's theorem, stating that a proper normal variety over an uncountable
algebraically closed field of characteristic 0 is determined by its category of
constructible abelian \'etale sheaves.
  We also discuss a conjecture in arbitrary characteristic, relating the
Zariski topological space to the perfection of a proper normal variety.
