Hajnal, András; Milner, E. C. On \(k\)-independent subsets of a closure. (English) Zbl 0788.04002 Stud. Sci. Math. Hung. 26, No. 4, 467-470 (1991). The authors continue the work of the second author and M. Pouzet [Algebra Univers. 21, 25-32 (1985; Zbl 0588.06004)] on independent subsets of closures on sets, generalizing earlier results concerning closures on ordered sets. They prove the following theorem: If \(\lambda\) is a singular strong limit cardinal, and if \(\varphi\) is a closure relation on a set \(E\) with dimension \(\dim(\varphi)=\lambda\), then, for each \(k<\omega\), there is a \(k\)-independent subset of \(E\) having cardinality \(\text{cf}(\lambda)\). Cited in 1 Document MSC: 03E05 Other combinatorial set theory Keywords:closure operator; independent subsets of closures; singular strong limit cardinal; closure relation; dimension Citations:Zbl 0588.06004 PDFBibTeX XMLCite \textit{A. Hajnal} and \textit{E. C. Milner}, Stud. Sci. Math. Hung. 26, No. 4, 467--470 (1991; Zbl 0788.04002)