Uniform bounds on growth in o-minimal structures

一致局部弱 o-极小性与 $*$-局部弱 o-极小性

Uniform local weak o-minimality and $*$-local weak o-minimality

本文引入一致局部弱 o-极小性与 $*$-局部弱 o-极小性，并证明相关结构在一元 $*$-连续性性质下的局部单调性定理及可定义集维数公式。

Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind

We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having definable bounded multiplication being affine.

Functions definable in definably complete uniformly locally o-minimal structure of the second kind

We investigate continuous functions definable in a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure). We prove a variant of the Arzela-Ascoli theorem for uniformly continuous definable functions and the following assertion: Consider the parameterized function $f:C \times P \rightarrow M$ which is equi-continuous with respect to $P$. The projection image of the set at which $f$ is discontinuous to the parameter space $P$ is of dimension smaller than $\dim P$ when $C$ is closed and bounded. In addition, we …

关于弱o-极小理论中一致有限性的注记

A note on uniform finiteness in weakly o-minimal theories

本文研究了有序群的弱o-极小扩张中一致有限性的等价条件，并指出域扩张中一致有限性可能不成立。

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

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 defiably Baire in the course of the proof of the inequality.

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

We demonstrate the following uniform local definable cell decomposition theorem in this paper. Consider a structure $\mathcal M = (M, <,0,+, \ldots)$ elementarily equivalent to a locally o-minimal expansion of the group of reals $(\mathbb R, <,0,+)$. Let $\{A_λ\}_{λ\inΛ}$ be a finite family of definable subsets of $M^{m+n}$. There exist an open box $B$ in $M^n$ containing the origin and a finite partition of definable sets $M^m \times B = X_1 \cup \ldots \cup X_k$ such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and $X_i \cap A_λ= \emp…