Effective zero-dimensionality for computable metric spaces

TL;DR

This paper explores effective notions of zero-dimensionality in computable metric spaces, establishing equivalences and characterizations, including an effective retract approach under compactness conditions.

Contribution

It introduces effective definitions and characterizations of zero-dimensionality in computable metric spaces, extending classical dimension theory into the computable analysis framework.

Findings

Several effectivisations of zero-dimensionality are shown to be equivalent.

The characterisation extends to higher dimensions and closed shrinkings.

An effective retract characterization is proven under compactness conditions.

Abstract

We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that concerns covering dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to the space and a class of subspaces) are defined; these correspond to definitions of inductive and covering dimensions and a countable basis condition. Finally, an effective retract characterisation of zero-dimensionality is proven under an effective compactness condition. In one direction this uses a version of the construction of bilocated sets.