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Applications of descriptive set theory to topology and analysis


Solecki, Slawomir J. (1995) Applications of descriptive set theory to topology and analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/s2fn-6633.


In Chapter 1, we prove that for every family I of closed subsets of a Polish space each ∑^1_1 set can be covered by countably many members of I or else contains a nonempty ∏^0_2 set which cannot be covered by countably many members of I. We derive from it the general form of Hurewicz's theorem due to Kechris, Louveau, and Woodin, and a theorem of Feng on the open covering axiom. Also some well known theorems on finding "big" closed sets inside of "big" ∑^1_1 sets are consequeces of our result. Chapter 2 consists of a joint work with A.S. Kechris. We prove that given a σ-ideal I, the possibility of approximating each ∑^1_1 set by a Borel set modulo I is equivalent to a definable form of the countable chain condition. This answers a question of Mauldin. We also characterize the meager ideal on a Polish group G as the only translation invariant, ccc σ-ideal I on G such that each set from I is contained in an F_σ set from I. This partially verifies a conjecture of Kunen. In Chapter 3, we establish a theorem which gives sufficient conditions for a K_σ equivalence relation to continuously embed E_o. As a consequence of this result we show that no indecomposable continuum contains a Borel set which has precisely one point in common with each composant. This solves an old problem in the theory of continua. In Chapter 4, answering a question of A.S. Kechris, we prove that the Topological Vaught Conjecture holds for Polish groups admitting invariant metrics. We also answer a question of R.L.Sami by proving that there exist continuous actions of Polish abelian groups with non-Borel induced orbit eqivalence relations. Actually, we give a fully algebraic characterization of sequences of countable abelian groups (H_n) such that the group ∏_n H_n has a continuous action with non-Borel orbit equivalence relation. In Chapter 5, we give a characterization of local compactness for Polish abelian groups in terms of Haar null sets of Christensen: a Polish abelian group is locally compact if each family of mutually disjoint closed (or, equivalently, universally measurable) sets which are not Haar null is countable. This completes, in a sense Dougherty's solution to a problem of Christensen. We also consider the question of the possibility of approximating analytic by Borel sets modulo Haar null sets. Chapter 6 contains two dichotomy theorems for Baire class 1 functions: a Baire class 1 function can be decomposed into countably many continuous functions, or else it contains a function which is as complicated with respect to decompositions into continuous functions as any other Baire class 1 function; an analogous theorem is proved for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprans. Their proofs use effective descriptive set theory as well as infinite games. Some results on decompositions of Borel sets and functions on higher levels are also obtained.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:W. P. Carey & Co., Inc., Prize in Mathematics, 1995
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kechris, Alexander S.
Thesis Committee:
  • Unknown, Unknown
Defense Date:5 May 1995
Record Number:CaltechTHESIS:12012011-093702619
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6744
Deposited By: Dan Anguka
Deposited On:01 Dec 2011 18:54
Last Modified:19 Apr 2021 22:34

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