A universal characterization of the closed Euclidean interval

M.H. Escardo, Alex Simpson

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in and elementary topos with natural numbers object
Original languageEnglish
Title of host publicationLogic in Computer Science, 2001. Proceedings. 16th Annual IEEE Symposium on
Number of pages11
Publication statusPublished - 2001


  • category theory
  • process algebra
  • programming theory
  • set theory
  • Euclidean topology
  • arithmetic operations
  • bounded real line segments
  • category
  • closed Euclidean interval
  • computable functions
  • elementary topos
  • interval object
  • primitive recursion
  • topological spaces
  • universal characterization
  • Arithmetic
  • Computer science
  • Convergence
  • Equations
  • Informatics
  • Logic
  • Mechanical factors
  • Set theory
  • Testing
  • Topology


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