Strong computable type

TL;DR

This paper provides a theoretical framework for understanding and analyzing the property of computable type in compact sets, establishing equivalences, and offering new topological criteria and invariants for its study.

Contribution

It introduces a unified approach to computable type, proves the equivalence of different definitions, and develops topological invariants for analyzing strong computable type.

Findings

Strong computable type can be characterized by topological invariants.

The relativized version of computable type is more suitable for topological analysis.

New criteria for determining computable type based on descriptive complexity.

Abstract

A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazovi\'c and other authors established the property for many other sets, such as manifolds. In this article we propose a theoretical study of the notion of computable type, in order to improve our general understanding of this notion and to provide tools to prove or disprove this property. We first show that the definitions of computable type that were distinguished in the literature, involving metric spaces and Hausdorff spaces respectively, are actually equivalent. We argue that the stronger, relativized version of computable type, is better behaved and prone to topological analysis. We obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants, as well as purely topological criteria. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the null-homotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones.