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Constructive decidability of classical continuity

Published online by Cambridge University Press:  23 December 2014

MARTÍN ESCARDÓ
MARTÍN ESCARDÓ*
Affiliation:
School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom Email: m.escardo@cs.bham.ac.uk
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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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 7: Computing with Infinite Data: Topological and Logical Foundations Part 1 , October 2015 , pp. 1578 - 1589
Copyright
Copyright © Cambridge University Press 2014 

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