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：开集是数学（尤其是分析和拓扑学）的核心概念，其地位之重要鲜有其他概念能及。在绝大多数（若非全部）数学的计算化处理中，开集仅能通过其“编码”或“表示”被间接研究。本文探讨了在给定任意实数开集的情况下，计算其最常见表示（即可数个开区间的集合）的难度。我们在Kleene的高阶可计算性理论框架下展开研究，该理论历史上基于S1-S9范式，而作者近期为其建立了直观的lambda演算表述。我们证明了将开集转换为上述表示的“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、乌雷松引理和蒂茨扩张定理）之间存在诸多计算等价性。同时，我们将这些泛函与受控函数和有界变差函数的已知运算，以及限制在闭集上的勒贝格测度进行了比较。针对后者，我们得到了若干与主流数学定理相关的自然计算等价结果。