Contributions to Pointfree Topology and Apartness Spaces

• Main Author: Hedin, Anton
• Format: Doctoral Thesis
• Language: English
• Published: Uppsala universitet, Algebra, geometri och logik 2011
• Subjects: Constructive mathematics; General topology; Pointfree topology; Domain theory; Interval analysis; Apartness spaces; Mathematical logic; Matematisk logik
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-152068; http://nbn-resolving.de/urn:isbn:978-91-506-2219-5

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.