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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
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URI: https://cronfa.swan.ac.uk/Record/cronfa65626
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first_indexed 2024-02-09T15:12:49Z
last_indexed 2024-02-09T15:12:49Z
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spelling v2 65626 2024-02-09 Point Degree Spectra of Represented Spaces 17a56a78ec04e7fc47b7fe18394d7245 0000-0002-0173-3295 Arno Pauly Arno Pauly true false 2024-02-09 SCS 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. Journal Article Forum of Mathematics, Sigma 10 Cambridge University Press (CUP) 2050-5094 27 5 2022 2022-05-27 10.1017/fms.2022.7 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University Another institution paid the OA fee 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). 2024-04-04T12:42:22.3500370 2024-02-09T15:00:48.1059608 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Takayuki Kihara 1 Arno Pauly 0000-0002-0173-3295 2 65626__29535__c5e37b910b464f899a7f564be533ae6e.pdf point-degree-spectra-of-represented-spaces.pdf 2024-02-09T15:02:45.9626947 Output 521194 application/pdf Version of Record true © The Author(s), 2022. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence. true eng https://creativecommons.org/licenses/by/4.0/
title Point Degree Spectra of Represented Spaces
spellingShingle Point Degree Spectra of Represented Spaces
Arno Pauly
title_short Point Degree Spectra of Represented Spaces
title_full Point Degree Spectra of Represented Spaces
title_fullStr Point Degree Spectra of Represented Spaces
title_full_unstemmed Point Degree Spectra of Represented Spaces
title_sort Point Degree Spectra of Represented Spaces
author_id_str_mv 17a56a78ec04e7fc47b7fe18394d7245
author_id_fullname_str_mv 17a56a78ec04e7fc47b7fe18394d7245_***_Arno Pauly
author Arno Pauly
author2 Takayuki Kihara
Arno Pauly
format Journal article
container_title Forum of Mathematics, Sigma
container_volume 10
publishDate 2022
institution Swansea University
issn 2050-5094
doi_str_mv 10.1017/fms.2022.7
publisher Cambridge University Press (CUP)
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
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description 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.
published_date 2022-05-27T12:42:19Z
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score 11.01353

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