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DIMENSION INEQUALITY FOR A DEFINABLY COMPLETE UNIFORMLY LOCALLY O-MINIMAL STRUCTURE OF THE SECOND KIND

Part of: Spaces with richer structures Special properties Model theory

Published online by Cambridge University Press:  07 September 2020

MASATO FUJITA
MASATO FUJITA*
Affiliation:
DEPARTMENT OF LIBERAL ARTS, JAPAN COAST GUARD ACADEMY, 5-1 WAKABA-CHO, KURE, HIROSHIMA737-8512, JAPANE-mail: [email protected]
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Abstract

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality $\dim (f(X)) \leq \dim (X)$ in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than $\dim (X)$. We also show that the structure is definably Baire in the course of the proof of the inequality.

Keywords

uniformly locally o-minimal structure of the second kinddefinably Baire structuredefinably complete

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality
Secondary: 54F45: Dimension theory 54E52: Baire category, Baire spaces
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1654 - 1663
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Copyright
© The Association for Symbolic Logic 2020

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References

REFERENCES

Dolich, A., Miller, C., and Steinhorn, C.. Structure having o-minimal open core. Transactions of the American Mathematical Society, vol. 362 (2010), pp. 1371–1411.CrossRefGoogle Scholar
Fornasiero, A. and Servi, T.. Definably complete Baire structure. Fundamenta Mathematicae, vol. 209 (2010), pp. 215–241.CrossRefGoogle Scholar
Fujita, M., Uniform local definable cell decomposition for locally o-minimal expansion of the group of reals, preprint, 2020, arXiv:2008.03494v1.Google Scholar
Fujita, M., Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition. Annals of Pure and Applied Logic, vol. 171 (2020), p. 102756.CrossRefGoogle Scholar
Hieronymi, P., An analogue of the Baire category theorem, this Journal, vol. 78 (2013), pp. 207–213.Google Scholar
Miller, C., Expansions of dense linear orders with the intermediate value property, this Journal, vol. 66 (2001), pp. 1783–1790.Google Scholar
Toffalori, C. and Vozoris, K., Notes on local o-minimality. Mathematical Logic Quarterly, vol. 55 (2009), pp. 617–632.CrossRefGoogle Scholar
van den Dries, L., Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar