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Type-definable and invariant groups in o-minimal structures

  • Jana Maříková (a1)

Abstract

Let M be a big o-minimal structure and G a type-definable group in Mn. We show that G is a type-definable subset of a definable manifold in Mn that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result holds for the wider class of so-called invariant groups: each invariant group G in Mn has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as Mn.

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[1]Berarducci, A. and Otero, M., Intersection theory for o-minimal manifolds. Annals of Pure and Applied Logic, vol. 107 (2001), no. 1-3, pp. 87–119.
[2]Bourbaki, N., Elements of mathematics, general topology, vol. 293, Hermann, Paris.
[3]Peterzil, Y. and Starchenko, S., Definable homomorphisms of abelian groups in o-minimal structures, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 1–27.
[4]Pillay, A., On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.
[5]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

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