## 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 Σ^{1}_{1} sets holds in the context of smooth sets. We also show
that the collection of Σ^{1}_{1} smooth sets is ∏^{1}_{1} on the codes. The analogs of
thin sets are called sparse sets. We prove that there is a largest ∏^{1}_{1} sparse set
and we give a characterization of it. We show that in L there is a ∏^{1}_{1} 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 ∏^{1}_{1} 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 Σ^{1}_{1} sets), then there is a largest ∏^{1}_{1} set in I^{int} (i.e., every closed
subset of it is in I). For σ-ideals on 2^{ω} we present a characterization of this
set in a similar way as for C_{1}, the largest thin ∏^{1}_{1} set. As a corollary we get
that if there are only countable many reals in L, then the covering property
holds for Σ^{1}_{2} 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 | ||||||||

Research Advisor(s): | - Kechris, Alexander S.
| ||||||||

Thesis Committee: | - Unknown, Unknown
| ||||||||

Defense Date: | 16 May 1990 | ||||||||

Funders: |
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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 | ||||||||

Last Modified: | 18 Mar 2015 20:25 |

## Thesis Files

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