ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

Kihara, Taro

doi:10.1017/jsl.2019.18

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2022

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“…This has already proven to be a rich source for the fine-grained study of the enumeration degrees, as both previously studied substructures as well as new ones of interest to computability theorists appear in this fashion. Kihara explored the truth-table reducibility variant of our generalised Turing degrees in [28].…”

Section: Introductionmentioning

confidence: 99%

Point Degree Spectra of Represented Spaces

Kihara

Pauly

2022

Forum of Mathematics, Sigma

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We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

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Section: Introductionmentioning

confidence: 99%