Co-c.e. spheres and cells in computable metric spaces

TL;DR

This paper explores conditions in computable metric spaces under which co-computably enumerable spheres and cells are guaranteed to be computable, utilizing higher-dimensional chains and spherical chains.

Contribution

It establishes that in locally computable spaces, co-c.e. spheres and cells with co-c.e. boundaries are computable, advancing understanding of computability in metric spaces.

Findings

Co-c.e. spheres are computable in locally computable spaces.

Co-c.e. cells with co-c.e. boundary spheres are computable.

Higher-dimensional chains are used to prove these results.

Abstract

We investigate conditions under which a co-computably enumerable set in a computable metric space is computable. Using higher-dimensional chains and spherical chains we prove that in each computable metric space which is locally computable each co-computably enumerable sphere is computable and each co-c.e. cell with co-c.e. boundary sphere is computable.