Book ChapterDOI
Completely Regular Bishop Spaces
Iosif Petrakis
- pp 302-312
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TLDR
The quotient, the pointwise exponential and the completely regular Bishop spaces are introduced, including a generalized version of the Tychonoff embedding theorem for Bishop spaces.Abstract:
Bishop’s notion of a function space, here called a Bishop space, is a constructive function-theoretic analogue to the classical set-theoretic notion of a topological space. Here we introduce the quotient, the pointwise exponential and the completely regular Bishop spaces. For the latter we present results which show their correspondence to the completely regular topological spaces, including a generalized version of the Tychonoff embedding theorem for Bishop spaces. All our proofs are within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).read more
Citations
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Constructive topology of bishop spaces
TL;DR: The theory of Bishop spaces (TBS) as mentioned in this paper is the most recent approach to constructive topology with points, and it has been studied extensively in the literature, e.g., in the context of constructive point-function topologies.
Book ChapterDOI
The Urysohn Extension Theorem for Bishop Spaces
TL;DR: The main result is the translation within the theory of Bishop spaces of the Urysohn extension theorem, which it is shown that it is constructively provable.
Proceedings ArticleDOI
A constructive function-theoretic approach to topological compactness
TL;DR: 2-compactness is introduced, a constructive function-theoretic alternative to topological compactness, based on the notions of Bishop space and Bishop morphism, and the countable Tychonoff compactness theorem is proved for them.
Journal ArticleDOI
Constructive uniformities of pseudometrics and Bishop topologies
TL;DR: A ``large'' version of the Tychonoff embedding theorem for f-uniform spaces is proved and it is shown that the notion of morphism between uniform spaces captures Bishop continuity.
Journal ArticleDOI
Embeddings of Bishop spaces
TL;DR: The classical Urysohn extension theorem is translated within the theory of Bishop spaces, a constructive approach to point-function topology and a natural constructive alternative to the classical theory of the rings of continuous functions.
References
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Journal ArticleDOI
Rings of Continuous Functions.
Book
Foundations of Constructive Analysis
TL;DR: In this article, the authors have no associated abstract for their paper, but they fix it (fix it) by using the following abstracts from the abstracts of the paper:
Journal ArticleDOI
Constructive Set Theory
TL;DR: There is a widespread current impression that the theory of Godel functionals, with quantifiers and choice, is the appropriate formalism for Bishop's book [1], but this is not so, and in more advanced mathematics the complexities become intolerable.
Journal ArticleDOI
Constructive Mathematics in Theory and Programming Practice
Douglas S. Bridges,Steeve Reeves +1 more
TL;DR: A survey of constructive mathematics can be found in this paper, where a sketch of both Myhill's axiomatic system for BISH and a constructive development of the real line R is given.