Coinitial families of perfect sets

Marek Balcerzak and Andrzej Roslanowski

We look at the structure of the family of perfect sets from an ideal I on the real line ${\mathbb R}$. We restrict our attention to ideals on ${\mathbb R}$ which are dense on perfect sets (i.e. each perfect set contains a perfect subset from the ideal). To compare different ideals from the point of view of their perfect members we use perfect isomorphisms. A bijection f from ${\mathbb R}$ onto ${\mathbb R}$ is a perfect isomorphism if and only if it is measurable with respect to the field of Marczewski's sets. We describe canonical perfect isomorphisms of ${\mathbb R}$ and apply this description to build some examples. Further we look at intersections of families of ideals and their traces on perfect sets.

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