Description:  This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the LaczkovichKomjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the LaczkovichKomjáth property, extending a theorem of Balcerzak and Głąb. 

Creator(s):  Kieftenbeld, Vincent 
Creation Date:  May 2010 
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UNT Libraries

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UNT Theses and Dissertations

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Publisher Name: University of North Texas
Publisher Info: Web: www.unt.edu
Place of Publication: Denton, Texas


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Description:  This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the LaczkovichKomjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the LaczkovichKomjáth property, extending a theorem of Balcerzak and Głąb. 

Degree: 
Name:
Doctor of Philosophy
Level:
Doctoral
Discipline:
Mathematics
Department:
Department of Mathematics
Grantor:
University of North Texas


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Subject(s): 


Keyword(s):  coanalytic equivalence relations  resolvable maps  complete metrizability  ordinal topologies  
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Partner: 
UNT Libraries


Collection: 
UNT Theses and Dissertations


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Resource Type:  Thesis or Dissertation  
Format:  Text  
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Holder:
Kieftenbeld, Vincent
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Copyright is held by the author, unless otherwise noted. All rights reserved.
