# Topology of Injective Endomorphisms of Real Algebraic Sets

Abstract : Using 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.
Type de document :
Article dans une revue
Mathematische Annalen, Springer Verlag, 2004, 328, pp.353-372

https://hal.archives-ouvertes.fr/hal-00128201
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Soumis le : mercredi 31 janvier 2007 - 10:25:43
Dernière modification le : lundi 5 février 2018 - 15:00:03

### Citation

Adam Parusinski. Topology of Injective Endomorphisms of Real Algebraic Sets. Mathematische Annalen, Springer Verlag, 2004, 328, pp.353-372. 〈hal-00128201〉

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