Mathematische Annalen

, Volume 328, Issue 1–2, pp 353–372 | Cite as

Topology of injective endomorphisms of real algebraic sets

  • Adam ParusińskiEmail author


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.


Equivalence Class Partial Order Local Property Topological Property Injective Morphism 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Département de MathématiquesU.M.RAngers CedexFrance

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