Nowhere dense set mappings on the generalized linear continua

Abstract

A set $A \subseteq X$ is free for a set mapping F:X→P(X) provided x∉F(y) for any distinct x, y in A. If F maps the reals R into nowhere dense subsets of R, then Bagemihl proved that there is an everywhere dense free set for F, and assuming CH Hechler showed that such an F does not always admit an uncountable free set. In this paper, we show that Bagemihl's theorem cannot be generalized to the generalized linear continua C _α for arbitrarily large ordinal α, and under GCH we extend Hechler's theorem to C _α for every α.