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.

翻译：开集在数学，特别是分析和拓扑学中，其核心地位鲜有其他概念能及。在大多数（即便不是全部）数学的计算方法中，开集仅通过其“编码”或“表示”被间接研究。本文研究给定任意实数开集时，计算其最常见表示（即可数开区间族）的难度。我们采用克莱尼高阶可计算性理论，该理论历史上基于S1-S9方案，如今已由作者赋予直观的lambda演算表述。我们建立了将开集转换为前述表示的“结构”泛函与主流数学中产生的泛函（如半连续函数的基本性质、乌雷松引理、蒂策扩张定理）之间的众多计算等价性。还将这些泛函与正则函数和有界变差函数的已知操作，以及限制于闭集的勒贝格测度进行比较。针对后者，我们获得了若干涉及主流数学定理的自然计算等价性。