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.
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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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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