论开集的若干计算性质
On some computational properties of open sets
AI总结：
该文在Kleene高阶可计算性理论框架下，研究实数开集到可数开区间集表示的计算难度，建立了该转换泛函与半连续函数性质、Urysohn引理等主流数学泛函及测度相关运算的多组计算等价关系。
摘要：
开集在数学尤其是分析学和拓扑学中占据核心地位，其重要性鲜有概念能及。在大多数（即便不是全部）数学计算研究路径中，人们仅通过开集的“编码”或“表示”对其进行间接研究。本文研究了给定任意实数开集时，计算其最常见表示（即可数开区间集）的难度。我们采用Kleene高阶可计算性理论开展研究，该理论历史上基于S1-S9概形，目前经本文作者研究已具备直观的λ演算表述形式。我们建立了多组计算等价关系：一边是将开集转换为上述表示的“结构”泛函，另一边是主流数学中衍生的泛函，例如半连续函数的基本性质、Urysohn引理以及Tietze扩张定理对应的泛函。我们还将这些泛函与已有的正则函数、有界变差函数运算，以及限制在闭集上的Lebesgue测度相关运算进行了比较，针对后者得到了一系列涉及主流数学定理的自然计算等价关系。
英文摘要：
Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene's higher-order computability theory, which was historically based on the S1-S9 schemes and which now has an intuitive lambda calculus formulation due to the authors. We establish many computational equivalences between on one hand the 'structure' functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma, and the Tietze extension theorem. We also compare these functionals to known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics.
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。
