Abstract
Topology has been shown to be definable in several cryptomorphically equivalent ways: by a neighborhood system, by a collection of open sets (be these given as a vector along the powerset or as a partial diagonal on it), by a collection of closed sets, or by a mapping to open kernels.
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
Notes
- 1.
Georg Aumann (1906–1980) was a professor at TU München since 1960. Already in 1934/35 he visited the Institute for Advanced Studies in Princeton as a Rockefeller scholar. Some consider him as one of the more significant mathematicians of the first half of the twentieth century, not least because of his book Reelle Funktionen, [Aum69]. The first author has in 1968 been with him among those who formally founded the Mathematics unit of TUM—terminating its existence as an informal substructure of the old faculty of ‘Allgemeine Wissenschaften’.
- 2.
One has a rather firm feeling for negation; e.g. monotony when doubly negated. Do we have a corresponding feeling for “∕” and “∖” and how they operate together? Earlier denotations “
” (once designed contrasting to “;”), “⋅ .”, and “. ⋅” (in diverging intention!) have provided some confusion as it has been reported already in [SS89, SS93].
” (once designed contrasting to “;”), “⋅ .”, and “. ⋅” (in diverging intention!) have provided some confusion as it has been reported already in [References
Samson Adepoju Adeleke and Peter M. Neumann. Relations Related to Betweenness: Their Structure and Automorphisms. Number 623 in Memoirs of the American Mathematical Society. American Mathematical Society, 1998.
Georg Aumann. Reelle Funktionen, volume 68 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1969. 2nd Edition.
Georg Aumann. Kontakt-Relationen. Sitzungsberichte der Bayer. Akademie der Wissenschaften, Math.-Nat. Klasse, 1970.
Georg Aumann. AD ARTEM ULTIMAM — Eine Einführung in die Gedankenwelt der Mathematik. R. Oldenbourg München Wien, 1974. ISBN 3-486-34481-1.
Gunther Schmidt. Relational Mathematics, volume 132 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2011. ISBN 978-0-521-76268-7, 584 pages.
Gunther Schmidt and Thomas Ströhlein. Relationen und Graphen. Mathematik für Informatiker. Springer-Verlag, 1989. ISBN 3-540-50304-8, ISBN 0-387-50304-8.
Gunther Schmidt and Thomas Ströhlein. Relations and Graphs — Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993. ISBN 3-540-56254-0, ISBN 0-387-56254-0.
M. L. J. van de Vel. Theory of Convex Structures, volume 50 of North-Holland Mathematical Library. North-Holland, 1993.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Schmidt, G., Winter, M. (2018). Closures and Their Aumann Contacts. In: Relational Topology. Lecture Notes in Mathematics, vol 2208. Springer, Cham. https://doi.org/10.1007/978-3-319-74451-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-74451-3_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74450-6
Online ISBN: 978-3-319-74451-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)