## 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)

### MSC:

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

### Keywords:

independent functions; perfect set
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