关于若尔当曲线的构造性理论
On the Constructive Theory of Jordan Curves
AI总结：
在Bishop构造性分析框架下，证明Dieudonné式若尔当曲线定义与Berg等人的定义等价，建立其指标函数性质，并证明分段光滑若尔当曲线的指标与卷绕数一致，含新结果与说明性内容。
摘要：
我们使用类似于Dieudonné的若尔当曲线定义，证明了该概念与Berg等人在构造性证明若尔当曲线定理时所用的概念等价。随后，我们建立了若尔当曲线及其对应指标函数的若干性质，包括重要的命题32及其关于直线在光滑点穿过若尔当曲线的推论。最后一节致力于证明复平面中分段光滑若尔当曲线关于某点的指标与该曲线绕该点的熟知卷绕数相同。本文在Bishop构造性分析的框架下撰写，尽管第3-5节的工作几乎完全是新的，但为方便读者，文中包含了大量说明性材料。
英文摘要：
Using a definition of Jordan curve similar to that of Dieudonné, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of Jordan curves and their corresponding index functions, including the important Proposition 32 and its corollaries about lines crossing a Jordan curve at a smooth point. The final section is dedicated to proving that the index of a point with respect to a piecewise smooth Jordan curve in the complex plane is identical to the familiar winding number of the curve around that point. The paper is written within the framework of Bishop's constructive analysis. Although the work in Sections 3--5 is almost entirely new, the paper contains a substantial amount of expository material for the benefit of the reader.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
