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Tukey quotients, pre-ideals, and neighborhood filters with calibre (omega 1, omega)

Morgan, Jeremiah (2016) Tukey quotients, pre-ideals, and neighborhood filters with calibre (omega 1, omega). Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

This work seeks to extract topological information from the order-properties of certain pre-ideals and pre-filters associated with topological spaces. In particular, we investigate the neighborhood filter of a subset of a space, the pre-ideal of all compact subsets of a space, and the ideal of all locally finite subcollections of an open cover of a space. The class of directed sets with calibre (omega 1, omega) (i.e. those whose uncountable subsets each contain an infinite subset with an upper bound) play a crucial role throughout our results. For example, we prove two optimal generalizations of Schneider's classic theorem that a compact space with a G_delta diagonal is metrizable. The first of these can be stated as: if X is (countably) compact and the neighborhood filter of the diagonal in X^2 has calibre (omega 1, omega) with respect to reverse set inclusion, then X is metrizable. Tukey quotients are used extensively and provide a unifying language for expressing many of the concepts studied here.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Morgan, Jeremiahjeremiahsmorgan@gmail.comJSM63
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairGartside, Paulpmg20@pitt.eduPMG20
Committee MemberLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberHeath, Robertrwheath@pitt.eduRWHEATH
Committee MemberNyikos, Peternyikos@math.sc.edu
Date: 3 October 2016
Date Type: Publication
Defense Date: 21 June 2016
Approval Date: 3 October 2016
Submission Date: 29 July 2016
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 161
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: topology, directed sets, calibres, Tukey quotients, compact covers, P-paracompactness, metrizability, productivity, Lindelof Sigma-spaces, neighborhood filters, strong Pytkeev property, function spaces
Date Deposited: 03 Oct 2016 12:52
Last Modified: 15 Nov 2016 14:33
URI: http://d-scholarship.pitt.edu/id/eprint/28129

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