Abstract
As well known in a closure space \({(M, \mathfrak{D})}\) satisfying the exchange axiom and the finiteness condition we can complete each independent subset of a generating set of M to a basis of M (Theorem A) and any two bases have the same cardinality (Theorem B) (cf. [1,3,4,7]). In this paper we consider closure spaces of finite type which need not satisfy the finiteness condition but a weaker condition (cf. Theorem 3.5). We prove Theorems A and B for a closure space of finite type satisfying a stronger exchange axiom. An example is given satisfying this strong exchange axiom, but not Theorems A and B.
Similar content being viewed by others
References
Birkhoff G.: Lattice Theory. A.M.S. Colloq. Publ., Providence (1967)
Buekenhout F.: Espaces à fermeture. Bull. Soc. Math. Belg. 19, 147–178 (1967)
Cohn P.M.: Universal Algebra. D. Reidel, Dordrecht (1981)
Delandtsheer, A.: Dimensional linear spaces. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, chapt. 6. Elsevier, Amsterdam (1995)
Karzel H., Sörensen K., Windelberg D.: Einführung in die Geometrie. UTB Vandenhoeck, Göttingen (1973)
Kreuzer A., Sörensen K.: Exchange properties in closure spaces. J. Geom. 98, 127–138 (2010)
Schmidt, J.: Einige grundlegende Begriffe und Sätze aus der Theorie der Hüllenoperatoren. Ber. Math. Tagung Berl. 21–48 (1953)
Tukey J.-W.: Convergence and Uniformity in Topology. Princeton University Press, Princeton (1940)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Heinrich Wefelscheid on the occasion of his 70th birthday
Rights and permissions
About this article
Cite this article
Kreuzer, A., Sörensen, K. Closure Spaces of Finite Type. Results. Math. 59, 349–358 (2011). https://doi.org/10.1007/s00025-011-0104-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-011-0104-2