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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References
Johnson, W.: Topologizing Interpretable Sets in O-Minimal Structures. Unpublished note (2014)
Marker, D.: Model theory: An introduction, Graduate Texts in Mathematics, vol. 217. Springer, New York (2002)
Onshuus, A., Steinhorn, C.: On linearly ordered structures of finite rank. J. Math. Log. 9(2), 201–239 (2009)
Peterzil, Y., Steinhorn, C.: Definable compactness and definable subgroups of o-minimal groups. J. Lond. Math. Soc. (2) 59(3), 769–786 (1999)
Pillay, A.: First order topological structures and theories. J. Symb. Log. 52(3), 763–778 (1987)
van den Dries, L.: Tame Topology and o-Minimal Structures. London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)
Walsberg, E.: On the Topology of Metric Spaces Definable in o-Minimal Expansions of Fields (2015). Preprint. arXiv:1510.07291
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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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DOI: https://doi.org/10.1007/s00153-019-00680-z