On the constructive notion of closure maps
type: Item
instance of: scholarly article
title: On the constructive notion of closure maps
zbMATH Open document ID: 1259.03078
author: Mohammad Ardeshir
author: Rasoul Ramezanian
publication date: 2012-09-12T00:00:00Z
Mathematics Subject Classification ID: 03F60
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 26E40
zbMATH DE Number: 6081354
zbMATH Keywords: constructive reverse mathematics
zbMATH Keywords: closure maps
zbMATH Keywords: fan theorem
MaRDI profile type: Publication
full work available at URL: malq.201110040
OpenAlex ID: W2040777999
cites work: Decidability and Specker sequences in intuitionistic mathematics
cites work: The double negation of the intermediate value theorem
cites work: A fan-theoretic equivalent of the antithesis of Specker's theorem
cites work: The anti-Specker property, a Heine-Borel property, and uniform continuity
cites work: Q4929338
cites work: Constructive notions of equicontinuity
cites work: A constructive treatment of Urysohn's Lemma in an apartness space
cites work: Glueing continuous functions constructively
cites work: Sequences of real functions on [0,1] in constructive reverse mathematics
cites work: Constructivism in mathematics. An introduction. Volume I
cites work: An interpretation of intuitionistic analysis
cites work: Q5310896
DOI: 10.1002/MALQ.201110040
published in: Mathematical Logic Quarterly (MLQ)
review text: This paper is a contribution to the programme of constructive reverse mathematics, carried out in Bishop's constructive mathematics.NEWLINENEWLINEThe authors show that, under the assumption of certain additional intuitionistic principles, there exists no closure map such that both (1) all nontrivial open sets of \(\mathbb R\) are strictly included in their closure; and (2) the closure of a set \(A\) is the smallest functionally closed set of which \(A\) is a subset.NEWLINENEWLINEThroughout this paper, the authors introduce many additional principles that can be consistently added to Bishop's constructive mathematics, and they prove many positive and negative results on the notions of closed sets and closure maps. The negative results are obtained by way of Beth models.
reviewed by: Iris Loeb
description: scientific article; zbMATH DE number 6081354