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：开集在数学中（尤其是在分析和拓扑学中）占据核心地位，其重要性鲜有其他概念可比。在大多数（若非全部）计算数学方法中，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，计算其最常用表示（即可数开区间集）的难度。我们基于克林高阶可计算性理论展开研究，该理论历史上源于S1-S9方案，目前作者已为其建立了直观的λ演算表述。我们建立了大量计算等价关系：一方面是将开集转换为上述表示的“结构”泛函，另一方面是主流数学中产生的泛函，例如半连续函数的基本性质、乌拉松引理以及蒂策扩张定理。我们还将这些泛函与有界变差函数和正则函数的已知运算，以及限制于闭集上的勒贝格测度进行比较。针对后者，我们获得了源于主流数学定理的若干自然计算等价关系。