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Definable one-dimensional topologies in O-minimal structures

Abstract

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space \(\left( X,\tau \right) \) is definably homeomorphic to an affine definable space (namely, a definable subset of \(M^{n}\) with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

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Correspondence to Ya’acov Peterzil.

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Peterzil, Y., Rosel, A. Definable one-dimensional topologies in O-minimal structures. Arch. Math. Logic 59, 103–125 (2020). https://doi.org/10.1007/s00153-019-00680-z

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Keywords

  • o-minimality
  • Definable topology
  • One-dimensional topology

Mathematics Subject Classification

  • 03C64