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On the uniqueness of Polish group topologies

Pejic, Bojana (2007) On the uniqueness of Polish group topologies. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Polish groups are separable completely metrizable topological groups. A key problem in the theory of Polish groups is that of the uniqueness of a Polish group topology: under what conditions does a group admit only one Polish group topology? Closely related is the problem of automatic continuity: when is a homomorphism between Polish groups necessarily continuous? This dissertation is an investigation of these questions.The key to unlocking these problems is to determine which sets in a Polish group are definable both algebraically and topologically. By algebraically definable one has in mind sets such as the commutators, conjugacy classes, or the squares. In the context of Polish groups, topologically definable means being a Borel set. A classical uniqueness result requires algebraically definable sets that are always Borel. Unfortunately its use is sometimes limited: while algebraically definable sets are often analytic (continuous images of Borel sets), it is shown here that they are not necessarily Borel. In particular, the set of squares in the homeomorphism group of the unit circle, and the set of squares in the automorphism group of the rational circle, are not Borel.An alternative result that avoids the need for Borel sets is obtained: a Polish group with a neighborhood base at the identity consisting of sets that are always analytic has a unique Polishgroup topology. As a consequence of this result, compact, connected, simple Lie groups and finitely generated profinite groups have a unique Polish group topology.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Pejic, Bojanabop4@pitt.eduBOP4
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairGartside, Paulgartside@math.pitt.eduPMG20
Committee MemberLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberHeath, Robertrwheath@pitt.eduRWHEATH
Committee MemberUspenskiy,
Date: 26 September 2007
Date Type: Completion
Defense Date: 7 August 2007
Approval Date: 26 September 2007
Submission Date: 30 July 2007
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
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: complete analytic; Polish group topology; Borel; squares; Polish group
Other ID:, etd-07302007-174220
Date Deposited: 10 Nov 2011 19:55
Last Modified: 15 Nov 2016 13:47


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