Small opaque sets.

Abstract

A set in a separable metric space is called \emph{Borel--opaque} if it meets every Borel set of positive topological dimension. We show that if there is a set of reals with cardinality of the continuum and universal measure zero, then each separable space contains a Borel--opaque set that is of universal measure zero. Similar results hold for opaque sets that are perfectly meager, \la sets, \lap sets etc., and can be extended to some nonseparable spaces. On the other hand, we show that a \s set is zero--dimensional. Using opacity we also construct universal measure zero sets of positive Hausdorff dimension.