This thesis is divided into three chapters. In the first chapter we studythe smooth sets with respect to a Borel equivalence realtion E on a Polishspace X. The collection of smooth sets forms σ-ideal. We think of smoothsets as analogs of countable sets and we show that an analog of the perfectset theorem for Σ11 sets holds in the context of smooth sets. We also showthat the collection of Σ11 smooth sets is ∏11 on the codes. The analogs ofthin sets are called sparse sets. We prove that there is a largest ∏11 sparse setand we give a characterization of it. We show that in L there is a ∏11 sparseset which is not smooth. These results are analogs of the results known forthe ideal of countable sets, but it remains open to determine if large cardinalaxioms imply that ∏11 sparse sets are smooth. Some more specific results areproved for the case of a countable Borel equivalence relation. We also studyI(E), the σ-ideal of closed E-smooth sets. Among other things we prove thatE is smooth iff I(E) is Borel.
In chapter 2 we study σ-ideals of compact sets. We are interested in therelationship between some descriptive set theoretic properties like thinness,strong calibration and the covering property. We also study products of σ-idealsfrom the same point of view. In chapter 3 we show that if a σ-idealI has the covering property (which is an abstract version of the perfect settheorem for Σ11 sets), then there is a largest ∏11 set in Iint (i.e., every closedsubset of it is in I). For σ-ideals on 2ω we present a characterization of thisset in a similar way as for C1, the largest thin ∏11 set. As a corollary we getthat if there are only countable many reals in L, then the covering propertyholds for Σ12 sets.
