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.
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Dedicated to Heinrich Wefelscheid on the occasion of his 70th birthday
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Kreuzer, A., Sörensen, K. Closure Spaces of Finite Type. Results. Math. 59, 349–358 (2011). https://doi.org/10.1007/s00025-011-0104-2
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DOI: https://doi.org/10.1007/s00025-011-0104-2