保持像的连续局部可定义映射的可微逼近
Differentiable approximation of continuous locally definable maps that preserves the image
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中文总结 AI 辅助
本文将紧可定义集上连续可定义映射的像保持型$\boldsymbol{\textit{C}}^p$逼近结果，推广至局部紧局部可定义集上的连续局部可定义映射，结合o-极小与PL几何，运用去奇异性技术完成论证。
中文摘要 AI 辅助
近期，我们证明了定义在紧可定义集上的连续可定义映射可被$\boldsymbol{\textit{C}}^p$类连续可定义映射一致逼近，且不改变其像。本文旨在将该结果推广至连续局部可定义映射，同时考虑强Whitney拓扑，研究对象为定义在局部紧局部可定义集上的此类映射。论证结合了o-极小几何与PL几何，关键运用了Pawłucki的去奇异性技术及我们之前针对紧情形的结果。
英文摘要
Recently, we showed that continuous definable maps defined on compact definable sets can be uniformly approximated by continuous definable maps of class $\mathcal{C}^p$ without changing their image. The aim of this paper is to extend the previous result, this time taking into account the (strong) Whitney topology, to continuous locally definable maps defined on locally compact locally definable sets. The argument is an interplay between o-minimal and PL geometry and makes essential use of Pawłucki's desingularization techniques as well as our aforementioned result for the compact case.
