Abstract
In advanced functional programming, researchers have investigated the existential image, the power transpose, and the power relator, e.g. It will be shown how the existential image is of use when studying continuous mappings between different topologies relationally. Normally, structures are compared using homomorphisms and sometimes isomorphisms. This applies to group homomorphisms, to graph homomorphisms and many more. The technique of comparison for topological structures will be shown to be quite different. Having in mind the cryptomorphic versions of neighborhood topology, open kernel topology, open sets topology, etc., this seems important.
Lifting concepts to a relational and, thus, algebraically manipulable and shorthand form, shows that existential and inverse images must here be used for structure comparison. Applying the relational language TituRel to such topological concepts allows to study and also visualize them.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bird, R.S., de Moor, O.: Algebra of Programming. Prentice-Hall International (1996)
de Roever, W.-P., Engelhardt, K.: Data Refinement: Model-Oriented Proof Methods and their Comparison. Cambridge Tracts in Theoretical Computer Science, vol. 47. Cambridge University Press (1998)
Franz, W.: Topologie I. Sammlung Göschen 1181. Walter de Gruyter (1960)
Schmidt, G., Ströhlein, T.: Relationen und Graphen. Mathematik für Informatiker. Springer (1989) ISBN 3-540-50304-8, ISBN 0-387-50304-8
Schmidt, G., Ströhlein, T.: Relations and Graphs — Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer (1993) ISBN 3-540-56254-0, ISBN 0-387-56254-0
Schmidt, G.: Relational Language. Technical Report 2003-05, Fakultät für Informatik, Universität der Bundeswehr München, 101 pages (2003), http://mucob.dyndns.org:30531/~gs/Papers/LanguageProposal.html
Schmidt, G.: Relational Mathematics, Encyclopedia of Mathematics and its Applications, vol. 132, 584 p. Cambridge University Press (2011) ISBN 978-0-521-76268-7
Schmidt, G.: TituRel: Sprache für die Relationale Mathematik. Technical Report 132, Arbeitsberichte des Instituts für Wirtschaftsinformatik, Universität Münster, 11 pages (2011), http://mucob.dyndns.org:30531/~gs/Papers/Raesfeld2011ExtendedAbstract.pdf
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Schmidt, G. (2014). A Point-Free Relation-Algebraic Approach to General Topology. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2014. Lecture Notes in Computer Science, vol 8428. Springer, Cham. https://doi.org/10.1007/978-3-319-06251-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-06251-8_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06250-1
Online ISBN: 978-3-319-06251-8
eBook Packages: Computer ScienceComputer Science (R0)