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，具有核心地位，其重要性是少数概念所能比拟的。在数学的大多数（如果不是全部）计算方法中，开集仅通过其“代码”或“表示”被间接研究。本文研究给定任意实数开集时，计算其最常见的表示（即可数开区间集）的难度。我们基于Kleene的高阶可计算性理论展开工作，该理论历史上基于S1-S9方案，如今因作者的工作而具有一种直观的λ-演算形式。我们建立了许多计算等价性：一方面是将开集转换为前述表示的结构泛函，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、Urysohn引理和Tietze扩张定理。我们还将这些泛函与关于正则函数和有界变差函数的已知运算，以及限制在闭集上的Lebesgue测度进行了比较。针对后者，我们利用主流数学中的定理得到了若干自然的计算等价性结果。