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The Urysohn Extension Theorem for Bishop Spaces

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Logical Foundations of Computer Science (LFCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

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Abstract

Bishop’s notion of function space, here called Bishop space, is a function-theoretic analogue to the classical set-theoretic notion of topological space. Bishop introduced this concept in 1967, without exploring it, and Bridges revived the subject in 2012. The theory of Bishop spaces can be seen as a constructive version of the theory of the ring of continuous functions. In this paper we define various notions of embeddings of one Bishop space to another and develop their basic theory in parallel to the classical theory of embeddings of rings of continuous functions. Our main result is the translation within the theory of Bishop spaces of the Urysohn extension theorem, which we show that it is constructively provable. We work within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\), inductive definitions with countably many premises included.

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Notes

  1. 1.

    The uniqueness property is included, for example, in [21], p. 238.

  2. 2.

    According to Bishop and Bridges [4], p. 85, if \(B \subseteq X\), where (X, d) is an inhabited metric space, B is a bounded subset of X, if there is some \(x_{0} \in X\) such that \(B \cup \{x_{0}\}\) with the induced metric is a bounded metric space. If we suppose that the inclusion map of a subset is the identity (see [4], p. 68), the induced metric on \(B \cup \{x_{0}\}\) is reduced to the relative metric on \(B \cup \{x_{0}\}\). We may also denote a bounded subset B of an inhabited metric space X by \((B, x_{0}, M)\), where \(M > 0\) is a bound for \(B \cup \{x_{0}\}\). If \((B, x_{0}, M)\) is a bounded subset of X then \(B \subseteq \mathcal {B}(x_{0}, M)\), and \((\mathcal {B}(x_{0}, M), x_{0}, 2M)\) is also a bounded subset of X. I.e., a bounded subset of X is included in an inhabited bounded subset of X which is also metric-open i.e., it includes an open ball of every element of it.

  3. 3.

    See definition 2.1 in [4], p. 72. It is also easy to see that \(a \bowtie _{\mathbb {R}} b \leftrightarrow a \bowtie _{{\mathrm {Bic}({\mathbb R})}} b\), for every \(a, b \in \mathbb {R}\).

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Authors and Affiliations

  1. University of Munich, Theresienstr. 39, 80333, Munich, Germany

    Iosif Petrakis

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  1. Iosif Petrakis
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Correspondence to Iosif Petrakis .

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  1. City University of New York , New York, New York, USA

    Sergei Artemov

  2. Cornell University , Ithaca, New York, USA

    Anil Nerode

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Petrakis, I. (2016). The Urysohn Extension Theorem for Bishop Spaces. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_21

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  • DOI: https://doi.org/10.1007/978-3-319-27683-0_21

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