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
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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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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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