Approximation and zero set of definable functions in a definably complete locally o-minimal structure

2023/01/11 by Masato Fujita, Fujita, Masato, Tomohiro Kawakami +1
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2301.04264

Abstract

We consider a definably complete locally o-minimal expansion of an ordered field. We treat two topics in this paper. The first topic is a definable \mathcal C^r approximation of a definable \mathcal C^r-1 map between definable \mathcal C^r submanifolds in the definable \mathcal C^r-1 topology. The second topic is the imbedding theorem for definably compact definable \mathcal C^r manifolds. We demonstrate that a definably normal definable \mathcal C^r manifold is a definably \mathcal C^r diffeomorphic to a definable \mathcal C^r submanifold. It enables us to show that the definable quotient of a definably compact definable \mathcal C^r group by a definable subgroup exists.

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