  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.
