ANNALES POLONICI MATHEMATICI LXXVI.1-2 (2001) A characterization of proper regular mappings by T. Krasiński and S. Spodzieja (Łódź) Abstract. Let X, Y be complex affine varieties and f : X → Y a regular mapping. We prove that if dim X ≥ 2 and f is closed in the Zariski topology then f is proper in the classical topology. 1. Introduction. Let X , Y be complex affine varieties (i.e. irreducible algebraic subsets of complex linear spaces) and f : X → Y a regular map- ping. From the Constructibility Theorem of Chevalley ([Ł 2 ], VII.8.3, [M], Proposition 2.31) it easily follows that if f is proper in the classical topo- logy then f is closed in the Zariski topology (i.e. for any algebraic subset V of X the image f (V ) is an algebraic subset of Y ). In this paper we prove that the converse is true provided dim X ≥ 2 (cf. [RS] for polynomial mappings f : C n → C k ). Theorem 1.1. Let X , Y be complex affine varieties , dim X ≥ 2 and f : X → Y a non-constant regular mapping. If f is closed in the Zariski topology then f is proper in the classical topology. From this theorem and well known facts we obtain the following charac- terization of finite mappings for affine varieties: Corollary. Let X , Y be complex affine varieties , dim X ≥ 2 and f : X → Y a non-constant regular mapping. Then the following conditions are equivalent : (i) f is finite , i.e. C[X ] is integral over f ∗ (C[Y ]), (ii) f is proper in the classical topology , (iii) f is closed in the classical topology , (iv) f is closed in the Zariski topology. 2000 Mathematics Subject Classification : Primary 14R99; Secondary 32H35. Key words and phrases : polynomial mapping, Łojasiewicz exponent, proper mapping, algebraic set. This research was partially supported by KBN Grant No. 2 P03A 007 18. [127]