Constructive decidability of classical continuity

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Abstract

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

Details

Original languageEnglish
Pages (from-to)1578-1589
Number of pages12
JournalMathematical Structures in Computer Science
Volume25
Issue numberSpecial Issue 7
Early online date23 Dec 2014
Publication statusPublished - 1 Oct 2015