A generalization of Markov's approach to the continuity problem for Type 1 computable functions

TL;DR

This paper generalizes Markov's approach to the continuity problem for Type 1 computable functions, introducing new notions of effective closure and providing conditions that ensure computable functions are effectively continuous.

Contribution

It axiomatizes and extends Markov's method, defining various effective closure concepts and establishing conditions for effective continuity in computable topological spaces.

Findings

Effective closure notions prevent effective discontinuities.

Spaces with dense computable sequences satisfy effective continuity conditions.

Results unify and extend previous continuity theorems for computable functions.

Abstract

We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable functions are (effectively) continuous". We introduce different notions of effective closure. These notions of effective closure lead to different notions of effective discontinuity at a point. We give conditions that prevent computable functions from having effective discontinuities. We finally show that results that forbid effective discontinuities can be turned into (abstract) continuity results on spaces where the closure and effective closure of semi-decidable sets naturally coincide. This happens for instance on spaces which admit a dense and computable sequence.