Strong computable type

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, Iljazović 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.