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The Vitali covering theorem in constructive mathematics
FB 6: Mathematik, Universität Siegen, Siegen, Germany.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
(English)Article in journal (Refereed) Submitted
Abstract [en]

This paper investigates the Vitali Covering Theorem from various constructive angles. A Vitali Cover of a metric space is a cover such that for every point there exists an arbitrarily small set of the cover containing this point. The VCT now states, that for any Vitali Cover one can find a finite, disjoint family of sets in the Vitali Cover that cover the entire space up to a set of a given non-zero measure. We will show, by means of a recursive counterexample, that there cannot be a fully constructive proof, but that adding a very weak semi-constructive principle suffices to give such a proof. Lastly, we will show that with an appropriate formalization in formal topology the non-constructive problems can be avoided completely.

Keyword [en]
Constructive mathematics, Reverse mathematics, Measure theory, Vitali's covering theorem, Formal topology
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:uu:diva-152066OAI: oai:DiVA.org:uu-152066DiVA: diva2:412413
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
In thesis
1. Contributions to Pointfree Topology and Apartness Spaces
Open this publication in new window or tab >>Contributions to Pointfree Topology and Apartness Spaces
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2011. 40 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 71
Keyword
Constructive mathematics, General topology, Pointfree topology, Domain theory, Interval analysis, Apartness spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152068 (URN)978-91-506-2219-5 (ISBN)
Public defence
2011-06-08, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2011-05-17 Created: 2011-04-23 Last updated: 2011-06-14Bibliographically approved

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