https://hal.archives-ouvertes.fr/hal-00128201Parusinski, AdamAdamParusinskiLAREMA - Laboratoire Angevin de Recherche en MathÃ©matiques - UA - UniversitÃ© d'Angers - CNRS - Centre National de la Recherche ScientifiqueTopology of Injective Endomorphisms of Real Algebraic SetsHAL CCSD2004[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Arxiv, Import2007-01-31 10:25:432021-10-20 03:18:442007-01-31 10:25:43enJournal articles1Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduce that any injective regular self-mapping of a real algebraic set is a homeomorphism. We show also a similar local property. All our results can be extended to arc-symmetric semi-algebraic sets and injective continuous arc-symmetric morphisms, and some results to Euler semi-algebraic sets and injective continuous semi-algebraic morphisms.