Dimension inequality for a definably complete uniformly locally o-minimal structure of the second kind

Abstract

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X\to R^n$ be a definable map, where $X$ is a definable set and $R$ is the universe of the structure. We demonstrate the inequality $\dim (f(X))\le \dim (X)$ 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 defiably Baire in the course of the proof of the inequality.

References in corpus (1)

• Uniform local definable cell decomposition for locally o-minimal expansion of the group of reals

Cited by in corpus (8)

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• Tameness of definably complete locally o-minimal structures and definable bounded multiplication
• Almost o-minimal structures and $\mathfrak{X}$-structures
• Uniformly locally o-minimal open core
• Notes on definably complete locally o-minimal expansions of ordered groups
• Definable compactness in definably complete locally o-minimal structures
• Locally o-minimal open core
• Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind