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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: fujita.masato.p34@kyoto-u.jp
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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

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