On some computational properties of open sets

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.