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On some computational properties of open sets.

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Title: On some computational properties of open sets.
Authors: Normann, Dag1 (AUTHOR), Sanders, Sam2 (AUTHOR)
Source: Journal of Logic & Computation. Jun2025, Vol. 35 Issue 4, p1-38. 38p.
Subjects: Functions of bounded variation, Recursion theory, Lambda calculus, Lebesgue measure, Computational mathematics
Abstract: 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, in particular its equivalent lambda calculus formulation due to Platek. 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 with 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. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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, in particular its equivalent lambda calculus formulation due to Platek. 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 with 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/logcom/exae048
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      – Code: eng
        Text: English
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        PageCount: 38
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      – SubjectFull: Functions of bounded variation
        Type: general
      – SubjectFull: Recursion theory
        Type: general
      – SubjectFull: Lambda calculus
        Type: general
      – SubjectFull: Lebesgue measure
        Type: general
      – SubjectFull: Computational mathematics
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              Text: Jun2025
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              Y: 2025
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