Point Degree Spectra of Represented Spaces

Kihara, Taro; Pauly, Arno

doi:10.1017/fms.2022.7

Cited by 11 publications

(5 citation statements)

References 58 publications

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“…Formal relationship between these two problems can be obtained, topological conditions being equivalent to suitable relatizations of computabilitytheoretic conditions (more details can be found in [KP22,KNP19]). Let us just give more informal connections:…”

Section: Generically Weaker Topologymentioning

confidence: 99%

“…In order to study the problem of separating the notions of computable points associated to two topologies, we first relativize it, which yields a purely topological problem: whether two topologies are σ-homeomorphic, i.e., whether the space can be decomposed as a countable union of subsets such that the two topologies agree on each subset. The relationship between the computability-theoretic content of points and the σ-homeomorphism class of the space was thoroughly investigated by Kihara, Pauly and Ng [KP22,KNP19].…”

Section: Introductionmentioning

confidence: 99%

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Comparing Computability in Two Topologies

Amir¹,

Hoyrup²

2023

J. symb. log.

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Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points.We first study a purely topological version of the problem, which is to understand when two topologies are not σ-homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager w.r.t. a topology, and are computable w.r.t. another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, i.e. have a homeomorphic copy that is semicomputable but not computable.

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Section: Generically Weaker Topologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparing Computability in Two Topologies

Amir¹,

Hoyrup²

2023

J. symb. log.

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“…In a computable T 0 -space (X, τ ), a point can be identified with the set of its basic neighborhoods, therefore the computability properties of a point of such a space is entirely captured by the enumeration degree of its neighborhood basis. This idea is explored in depth by Kihara and Pauly in [KP14]. In particular, a computable reduction between points can be defined using enumeration reducibility as follows.…”

Section: Computablementioning

confidence: 99%

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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“…Levin's construction of a neutral measure [13] uses Sperner's lemma and was shown by Day and Miller [4] to also produce a nontotal continuous degree. More recently, Kihara and Pauly [12] and independently Hoyrup (unpublished) use facts from topological dimension theory to prove the existence of a nontotal continuous enumeration degree. The connection can be followed in the reverse direction as well.…”

mentioning

confidence: 99%

“…The connection can be followed in the reverse direction as well. For example, a structural property of the continuous enumeration degrees was the main tool in Kihara and Pauly's [12] solution to the second level Borel isomorphism problem; they constructed an uncountable Polish space which is neither second-level Borel isomorphic to the unit interval nor to the Hilbert cube.…”

mentioning

confidence: 99%