Miroslav Repick\'y
Perfect sets and collapsing continuum

Comment.Math.Univ.Carolinae 44,2 (2003) 315-327.

Abstract:Under Martin's axiom, collapsing of the continuum by Sacks forcing $\Bbb S$ is characterized by the additivity of Marczewski's ideal (see [4]). We show that the same characterization holds true if $\frak d=\frak c$ proving that under this hypothesis there are no small uncountable maximal antichains in $\Bbb S$. We also construct a partition of into $\frak c$ perfect sets which is a maximal antichain in $\Bbb S$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.

Keywords: Sacks forcing, Marczewski's ideal, cardinal invariants
AMS Subject Classification: Primary 03E40; Secondary 03E17