Complexity of families of compact sets in ℝ^<i>n</i>

Kovan-Bakan, Merve (2013) Complexity of families of compact sets in ℝ^<i>n</i>. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

The space of all compact subsets of ℝ^<i>n</i> with the Vietoris topology,
denoted <i><b>K</b></i>(ℝ^<i>n</i>), is a Polish space, i.e. separable and completely metrizable.
It is naturally stratified by dimension. In this work, we study the zero and one dimensional compact subsets of
ℝ^<i>n</i>, and two equivalence relations on <i><b>K</b></i>(ℝ^<i>n</i>):
the homeomorphism relation and the embedding relation induced by the action of autohomeomorphisms of
ℝ^<i>n</i>.

Among the zero dimensional compact subsets, Cantor sets are generic and form a Polish subspace.
We study the topological properties of this space as well as the structure with respect to the embedding relation.
Moreover, we show that the classification of Cantor sets up to embedding relation is at least as complex as the
classification of countable structures.

Next, we look into one dimensional compact subsets, particularly those that are connected, i.e. curves.
The curves also form a Polish subspace. We introduce a new connectedness property, namely strong arcwise connectedness.
We study the complexity of curves with this property using descriptive set theory tools, and show that the space of all
curves which are strong arcwise connected, is not Borel, and is exactly at the second level of the projective hierarchy.
In addition, we examine the classification of curves up to either equivalence relation and show that the curves are not
classifiable by countable structures.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Kovan-Bakan, Merve
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Gartside, Paul gartside@math.pitt.edu PMG20 Committee Member Darji, Udayan udayan.darji@louisville.edu Committee Member Espinoza, Benjamin bee1@pitt.edu BEE1 Committee Member Heath, Robert rwheath@pitt.edu RWHEATH Committee Member Lennard, Christopher lennard@pitt.edu LENNARD
Date:	26 February 2013
Date Type:	Publication
Defense Date:	2 August 2012
Approval Date:	26 February 2013
Submission Date:	30 July 2012
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	79
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	Cantor sets, embeddings, classification, turbulence, curves, Borel hierarchy, difference hierarchy, Projective hierarchy, strong arcwise connectedness, dendrites, dendroids
Date Deposited:	26 Feb 2013 17:54
Last Modified:	15 Nov 2016 14:06
URI:	http://d-scholarship.pitt.edu/id/eprint/16330

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