Nowhere dense set mappings on the generalized linear continua (Q803129)

If F maps a set X into its power set P(X) a subset \(A\subseteq X\) is said to be free if \(x\not\in F(Y)\) for any two distinct x, y in A. For infinite cardinals \((=\) initial ordinals) \(\alpha\) the author defines \(R_{\alpha}\) to be the set of all dyadic sequences of digits 0 and 1 of length \(\alpha\) which have a last digit 1. \(C_{\alpha}\) is the set of all dyadic sequences of length \(\alpha\) which contain at least one digit 1 and one digit 0 but which have no last digit 0. Both sets are linearly ordered by the lexicographic order and have the corresponding order topology. For this the author proves: 1) Let \(\alpha\) be a cardinal, \(\alpha '=cf \alpha\). Then there is a nowhere dense set mapping F on \(C_{\alpha}\) [that means: for every \(x\in C_{\alpha}\), F(x) is a nowhere dense subset of \(C_{\alpha}]\) such that the cardinality of any free set for F is at mot \(2^{\alpha '}\). (The case \(\alpha =\alpha '\) is trivial, so this statement is of interest only for singular \(\alpha\).) 2) If GCH is assumed, then there is a nowhere dense set mapping F on \(C_{\alpha}\) such that the cardinality of any free set for F is at most \(\alpha '\). This extends a theorem of \textit{S. H. Hechler} [Isr. J. Math. 11, 231-248 (1972; Zbl 0246.04003)].