# 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.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00128201
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Submitted on : Wednesday, January 31, 2007 - 10:25:43 AM
Last modification on : Monday, March 9, 2020 - 6:15:52 PM

### 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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