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# Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets

## Citation

Uzcátegui, Carlos (1990) Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03182015-110250011

## Abstract

This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ11 sets holds in the context of smooth sets. We also show that the collection of Σ11 smooth sets is ∏11 on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏11 sparse set and we give a characterization of it. We show that in L there is a ∏11 sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏11 sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ11 sets), then there is a largest ∏11 set in Iint (i.e., every closed subset of it is in I). For σ-ideals on 2ω we present a characterization of this set in a similar way as for C1, the largest thin ∏11 set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ12 sets.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Restricted to Caltech community only
• Kechris, Alexander S.
Thesis Committee:
• Unknown, Unknown
Defense Date:16 May 1990
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
NSFDMS-8718847
University of Los Andes (Venezuela)UNSPECIFIED
Record Number:CaltechTHESIS:03182015-110250011
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:03182015-110250011
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8783
Collection:CaltechTHESIS
Deposited By: Benjamin Perez
Deposited On:18 Mar 2015 20:25