Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.