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Point Degree Spectra of Represented Spaces

Takayuki Kihara, Arno Pauly Orcid Logo

Forum of Mathematics, Sigma, Volume: 10

Swansea University Author: Arno Pauly Orcid Logo

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DOI (Published version): 10.1017/fms.2022.7

Abstract

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, incl...

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Published in: Forum of Mathematics, Sigma
ISSN: 2050-5094
Published: Cambridge University Press (CUP) 2022
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa65626
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Abstract: 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.
College: Faculty of Science and Engineering
Funders: The work has benefitted from the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011-294962. For the duration of this research, the first author was partially supported by a Grant-in-Aid for JSPS fellows (FY2012–2014) and for JSPS overseas research fellows (FY2015–2016; Host: University of California, Berkeley).

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