网络与逆向数学，一项试点研究
Nets and Reverse Mathematics, a pilot study
摘要：
网是序列的概括，可能涉及不可数的索引集;这个概念是由摩尔和史密斯在大约一个世纪前提出的。他们还建立了概括网的各种基本定理的分析，由于博尔扎诺，维尔斯特拉斯，迪尼，Arzela，和其他人。最近，网是域理论发展的核心，提供了相关的斯科特和劳森拓扑的直观定义。本文用逆向数学的方法研究网的基本定理。我们限制自己的网索引的Baire空间的子集，因此三阶算术，因为这样的网足以获得我们的主要结果。在Kohlenbach的高阶反向数学基础理论之上，网的Bolzano-Weierstrass定理隐含了不可数覆盖的Heine-Borel定理。我们建立类似的结果，其他基本定理网，甚至一些等价，如迪尼的定理网。最后，我们证明了用序列替换网是困难的，但是用网替换序列可以避免对选择公理的需要，这是域理论中的一个基本问题。在附录中，我们研究了更一般的指数集的幂，建立了一个网的“大小”与相关收敛定理的幂成正比。
英文摘要：
Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano-Weierstrass, Dini, Arzela, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach's base theory of higher-order Reverse Mathematics, the Bolzano-Weierstrass theorem for nets implies the Heine-Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini's theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the 'size' of a net is directly proportional to the power of the associated convergence theorem.
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。
