Computability of pseudo-cubes

Horvat, Maja; Iljazović, Zvonko; Pažek, Bojan

doi:10.1016/j.apal.2020.102823

Cited by 5 publications

(2 citation statements)

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“…The first result can be used to prove that a pair is not in E n by approximating it with pairs that are not in E n in some way. It is taken from the definition of pseudo-cubes in [HIP20].…”

Section: Properties Of E(y )mentioning

confidence: 99%

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Strong computable type

Amir¹,

Hoyrup²

2022

Preprint

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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 nullhomotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones.

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“…The first result can be used to prove that a pair is not in E n by approximating it with pairs that are not in E n in some way. It is taken from the definition of pseudo-cubes in [HIP20].…”

Section: Properties Of E(y )mentioning

confidence: 99%

“…In [HIP20], the authors introduce the notion of a pseudo-n-cube, which is a pair (X, A) which can somehow be approximated by n-dimensional cubes and their boundaries. They prove that pseudo-n-cubes have computable type.…”

Section: Pseudo N-cubesmentioning

confidence: 99%