dc.contributor.advisor |
Mclntyre, D |
en |
dc.contributor.advisor |
Gauld, D |
en |
dc.contributor.author |
Richardson, Kerry Joseph |
en |
dc.date.accessioned |
2007-07-20T07:07:39Z |
en |
dc.date.available |
2007-07-20T07:07:39Z |
en |
dc.date.issued |
2000 |
en |
dc.identifier |
THESIS 00-517 |
en |
dc.identifier.citation |
Thesis (PhD--Mathematics)--University of Auckland, 2000 |
en |
dc.identifier.uri |
http://hdl.handle.net/2292/1019 |
en |
dc.description |
Full text is available to authenticated members of The University of Auckland only. |
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dc.description.abstract |
The topological "languages" referred to in the title are Resolutions and Scheduled Reflections. The constructive technique of resolutions uses the "language" of neighbourhood bases and are like a topological "microscope" in that they "reveal" greater detail and complexity of topological spaces. They also unify a vast array of seemingly desparate and ad hoc topological constructions. Scheduled Reflections use the "language" of elementary submodels and can handle inductive set-theoretic topological constructions. Both constructive techniques have been aimed by the author in concocting a counterexample to the conjectured large cardinal consistency result of the normal Moore space conjecture when the cardinality of the set of reals is small. "Small" here means not weakly inaccessible. Both techniques, resolutions and scheduled reflections, needed to be developed and while the first part of this thesis is devoted to various resolution constructions, scheduled reflections are not presented here. It turns out that scheduled reflections are probably more relevant to this possible counterexample than are resolutions.
Chapter 1 establishes some inner characterisations of general resolutions - a modification of Ul’janov resolutions. Chapter 2 gives inner characterisations for metrisable and discrete special resolutions. Chapter 3 identifies a limitation of special resolutions in relation to connectedness of topological spaces. We also show that some hyper-resolutions imply other notions of maps.
The second part of this thesis is a sketchbook that arose from a suggestion by Alan Dow. Alan, Frank Tall and William Weiss had proven the relative consistency with ZFC of, "every normal first countable space (and in particular every normal Moore space) is collectionwise normal", by means of a fusion of topology, supercompactness, and Cohen forcing. In the final model, c is a supercompact cardinal (although a model is also obtained where c is only a weakly compact cardinal).
Assuming large cardinals, is it consistent that every normal Moore space is metrisable when c = N2? Previous attempts to prove this only used the fact that Moore spaces are first countable. They also have a development and this could be utilised to obtain a proof. This alternative approach was not taken in this thesis.
Instead effort was focused on trying to show which partial orders are endowed in Chapter 5 and demonstrating how their existence would be helpful, after set-theoretic preliminaries in Chapter 4. We showed that one cannot get very far in iterating endowments. The attempted proofs of preservation results for endowments made no ground model assumptions; which turned out to be a handicap. Appendix 6 outlines the framework of the Dow, Tall, and Weiss paper.
Conversely, maybe c = N2 implies that there is a normal, non-metrisable Moore space. This might be shown by generalising Fleissner's CH example of such a space to c = N2. A scheduled reflection might be applied here for character reduction but is outside the scope of this present work. |
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dc.language.iso |
en |
en |
dc.publisher |
ResearchSpace@Auckland |
en |
dc.relation.ispartof |
PhD Thesis - University of Auckland |
en |
dc.relation.isreferencedby |
UoA9994049814002091 |
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dc.rights |
Restricted Item. Available to authenticated members of The University of Auckland. |
en |
dc.rights |
Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. |
en |
dc.rights.uri |
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm |
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dc.title |
Topological Languages and the Normal Moore Space Conjecture |
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dc.type |
Thesis |
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thesis.degree.grantor |
The University of Auckland |
en |
thesis.degree.level |
Doctoral |
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thesis.degree.name |
PhD |
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dc.rights.holder |
Copyright: The author |
en |
dc.identifier.wikidata |
Q111963689 |
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