Geodesic
Perfect sets and collapsing continuum
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 315-327 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $^\omega 2$ 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.
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 $^\omega 2$ 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.
Classification : 03E17, 03E40, 03E50, 54A35
Keywords: Sacks forcing; Marczewski's ideal; cardinal invariants 
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     author = {Repick\'y, Miroslav},
     title = {Perfect sets and collapsing continuum},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {315--327},
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Repický, Miroslav. Perfect sets and collapsing continuum. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 315-327. http://geodesic.mathdoc.fr/item/CMUC_2003_44_2_a9/