2020/10/06 by Masato Fujita, Fujita, Masato
Computer Science · Mathematics · #03C64 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2010.02420
openalex publication_date 2020/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 × P → 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 demonstrate that an archimedean DCULOAS structure which enjoys definable Tietze extension property is o-minimal. In the appendix, we show that an o-minimal expansion of an ordered group is not semi-bounded if and only if it enjoys definable Tietze extension property.