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Contributions to Pointfree Topology and Apartness Spaces
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
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 [en]
Constructive mathematics, General topology, Pointfree topology, Domain theory, Interval analysis, Apartness spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:uu:diva-152068ISBN: 978-91-506-2219-5OAI: oai:DiVA.org:uu-152068DiVA: diva2:412415
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
List of papers
1. Local Scott compactification
Open this publication in new window or tab >>Local Scott compactification
(English)Article in journal (Refereed) Submitted
Abstract [en]

We show how to embed certain formal topologies in locally Scott formal topologies. We call this process local Scott compactification. Examples include localic completions of metric spaces. We also prove a lifting result for morphisms. The local Scott compactification of a space corresponds to a domain representation of the space and in this way yields a space containing partial elements. In the case of the real numbers we obtain the space of partial reals, consisting of intervals where the endpoints are lower and upper reals respectively. We show that partial reals that define compact overt subspaces of the real numbers correspond precisely to intervals with real endpoints. The lifting of morphisms give sharp extensions of continuous real valued functions in the sense of interval analysis. These results are also generalized to normed vector spaces.

Keyword
Formal topologies, Continuous dcpos, Locally compact metric spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152064 (URN)
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
2. The domain theoretic derivative in formal topology
Open this publication in new window or tab >>The domain theoretic derivative in formal topology
(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
Formal topologies, Domains, Interval analysis, Differential calculus
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152065 (URN)
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-21Bibliographically approved
3. The Vitali covering theorem in constructive mathematics
Open this publication in new window or tab >>The Vitali covering theorem in constructive mathematics
(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
Constructive mathematics, Reverse mathematics, Measure theory, Vitali's covering theorem, Formal topology
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152066 (URN)
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
4. Towards set-presentable apartness spaces
Open this publication in new window or tab >>Towards set-presentable apartness spaces
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We investigate the theory of apartness spaces from a predicative point of view. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces. Moreover we discuss notions of set presentation for set-set apartness space.

Keyword
Constructive mathematics, General topology, Neighbourhood space, Apartness space
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152067 (URN)
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14

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