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The domain theoretic derivative in formal topology
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]

We investigate the possibility to develop constructively some of the theory of domain theoretical differential calculus by using formal topology. A formal point-free domain derivative of continuous functions on partial reals is defined and we prove that it is point-wise equal to the classical domain derivative.

Keyword [en]
Formal topologies, Domains, Interval analysis, Differential calculus
Research subject
Mathematical Logic
URN: urn:nbn:se:uu:diva-152065OAI: oai:DiVA.org:uu-152065DiVA: diva2:412412
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-21Bibliographically 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.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 71
Constructive mathematics, General topology, Pointfree topology, Domain theory, Interval analysis, Apartness spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
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)
Available from: 2011-05-17 Created: 2011-04-23 Last updated: 2011-06-14Bibliographically approved

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