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De Groot Duality for Represented Spaces

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Unity of Logic and Computation (CiE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13967))

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Abstract

We explore de Groot duality in the setting of represented spaces. The de Groot dual of a space is the space of closures of its singletons, with the representation inherited from the hyperspace of closed subsets. This yields an elegant duality, in particular between Hausdorff spaces and compact \(T_1\)-spaces. As an application of the concept, we study the point degree spectrum of the dual of Baire space, and show that it is, in a formal sense, far from being countably-based.

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Notes

  1. 1.

    The de Groot dual of a space is the same as the de Groot dual of its \(T_0\)-quotient anyway.

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Acknowledgements

We are grateful to Matthew de Brecht for fruitful discussions. We are also grateful to the anonymous referees for valuable suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematical Informatics, Nagoya University, Nagoya, Japan

    Takayuki Kihara & Arno Pauly

  2. School of Mathematics and Computer Science, Swansea University, Swansea, UK

    Takayuki Kihara & Arno Pauly

Authors
  1. Takayuki Kihara
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  2. Arno Pauly
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Corresponding author

Correspondence to Arno Pauly .

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Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Gianluca Della Vedova

  2. Kutaisi International University, Kutaisi, Georgia

    Besik Dundua

  3. University of Wisconsin, Madison, WI, USA

    Steffen Lempp

  4. University of Göttingen, Göttingen, Germany

    Florin Manea

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Kihara, T., Pauly, A. (2023). De Groot Duality for Represented Spaces. In: Della Vedova, G., Dundua, B., Lempp, S., Manea, F. (eds) Unity of Logic and Computation. CiE 2023. Lecture Notes in Computer Science, vol 13967. Springer, Cham. https://doi.org/10.1007/978-3-031-36978-0_8

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  • DOI: https://doi.org/10.1007/978-3-031-36978-0_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-36977-3

  • Online ISBN: 978-3-031-36978-0

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