Free sets for some special set mappings
Let f:X\(\to {\mathcal P}X\). We say \(A\subseteq X\) is free for f if \(x\not\in f(y)\) for all distinct x,y\(\in A\). Let \({\mathbb{R}}\) be the set of real numbers, and NWD the family of nowhere dense subsets of \({\mathbb{R}}\). The main result of the paper is the following. Assume Martin's Axiom. Given f:\({\mathbb{R}}\to NWD\), a sequence \(Q_ n(n\in \omega)\) of non-meager subsets of \({\mathbb{R}}\), and a cardinal \(\kappa <2^{\aleph_ 0}\), then there is \(A\subseteq {\mathbb{R}}\) free for f and such that \(| A\cap A_ n| =\kappa\) for \(n<\omega\). A number of similar results are obtained, mostly assuming Martin's Axiom. Solutions to the open problems raised by \textit{S. H. Hechler} [Isr. J. Math. 11, 213-248 (1972; Zbl 0246.04003)] are discussed.
scientific article; zbMATH DE number 4039906
scientific article