Perfect sets of independent functions. (English) Zbl 0823.54018

Let \({\mathbf k} = \{0, \dots, k-1\}\) be an integer \(\geq 2\), \(N\) the set of nonnegative integers. Then \({\mathbf k}^ N\) is the set of all mappings of \(N\) into \({\mathbf k}\). It is equipped with the product topology (and with this homeomorphic to Cantor’s discontinuum). If \(X \subseteq N\), \(C \subseteq {\mathbf k}^ N\) then \(C \upharpoonright X\) denotes the set of all restrictions \(f \upharpoonright X\) where \(f \in C\). Then the authors prove: There exists a perfect set \(C \subseteq {\mathbf k}^ N\) such that for each perfect set \(K \subseteq C\) there is an infinite set \(X \subseteq N\) for which \(K \upharpoonright X = {\mathbf k}^ X\). This theorem gives a positive answer to a question of M. Balcerzak [Some properties of ideals of sets in Polish spaces, Acta Univ. Lodz., Folia Math. (1991)]. Further it is proved: Let \(F \subseteq {\mathbf k}^ N\) be a perfect subset. Then there exists an infinite subset \(X \subseteq N\) such that \(F \upharpoonright X = {\mathbf k}^ X\) iff there exists a perfect independent subset of \(F\). Here a set \(F \subset {\mathbf k}^ X\) is called independent if for every sequence \(f_ 1, \dots, f_ n\) of different functions from \(F\) and every sequence \(x_ 1, \dots, x_ n\) of numbers from \({\mathbf k}\) the intersection \(f^{-1}_ 1 (x_ 1) \cap \dots \cap f^{-1}_ n (x_ n)\) is nonempty.
Reviewer: E.Harzheim (Köln)


54C99 Maps and general types of topological spaces defined by maps
03E20 Other classical set theory (including functions, relations, and set algebra)
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