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Conference on Computability in Europe

CiE 2019: Computing with Foresight and Industry pp 240–252Cite as

Borel and Baire Sets in Bishop Spaces

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

Abstract

We study the Borel sets \( \texttt {Borel} (F)\) and the Baire sets \( \texttt {Baire} (F)\) generated by a Bishop topology F on a set X. These are inductively defined sets of F-complemented subsets of X. Because of the constructive definition of \( \texttt {Borel} (F)\), and in contrast to classical topology, we show that \( \texttt {Baire} (F) = \texttt {Borel} (F)\). We define the uniform version of an F-complemented subset of X and we show the Urysohn lemma for them. We work within Bishop’s system \(\mathrm {BISH}^*\) of informal constructive mathematics that includes inductive definitions with rules of countably many premises.

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Notes

  1. 1.

    A different definition is given in [10]. See [19] for the relations between these two definitions.

  2. 2.

    Hence, if we define the set of Baire sets over an arbitrary family \(\varTheta \) of functions from X to \({\mathbb R}\), a sufficient condition so that \( \texttt {Baire} (\varTheta )\) is closed under complements is that \(\varTheta \) is closed under |.|, under wedge with \(\frac{1}{n}\) and under subtraction with \(\frac{1}{n}\), for every \(n \ge 1\). If \(\varTheta := \mathbb {F}(X, 2)\), then \(- \varvec{o}_{\mathbb {F}(X, 2)}(f) = \varvec{o}_{\mathbb {F}(X, 2)}(1-f) = \varvec{\zeta }_{\mathbb {F}(X, 2)}(f)\), hence by Proposition 4(ii) we get \( \texttt {Borel} (\mathbb {F}(X, 2)) = \texttt {Baire} (\mathbb {F}(X, 2))\).

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Acknowledgment

This paper was written during my research visit to CMU that was funded by the EU-research project “Computing with Infinite Data”. I would like to thank Wilfried Sieg for hosting me at CMU.

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Authors and Affiliations

  1. University of Munich, Munich, Germany

    Iosif Petrakis

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  1. Iosif Petrakis
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Correspondence to Iosif Petrakis .

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Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Petrakis, I. (2019). Borel and Baire Sets in Bishop Spaces. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_21

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_21

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