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(Unpublished)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
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Status: |
Unpublished |
Creators/Authors: |
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ETD Committee: |
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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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