Asymmetric Topology and Topological Spaces Defined By Games

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dc.contributor.author Cao, Jiling en
dc.date.accessioned 2007-06-25T21:50:33Z en
dc.date.available 2007-06-25T21:50:33Z en
dc.date.issued 1999 en
dc.identifier THESIS 99-236 en
dc.identifier.citation Thesis (PhD--Mathematics)--University of Auckland, 1999 en
dc.identifier.uri http://hdl.handle.net/2292/540 en
dc.description Full text is available to authenticated members of The University of Auckland only. en
dc.description.abstract In this thesis, we shall discuss various asymmetric topological structures, their relationships with topological games as well as their applications. The asymmetric structures we consider are quasi-metric spaces and quasi-uniform spaces and the classes of spaces which are "duals" of quasi-metric spaces. The concept of compactly symmetric quasi-uniform spaces is introduced, and its relationship with the notion of small-set symmetry due to Fletcher and Hunsaker is established. We find that both small-set and compact symmetries are well-behaved with hyperspaces and very useful to function spaces. Actually, small-set and compact symmetries enable us to reconcile the relations of the Vietoris topology and the Bourbaki quasi-uniformity; and they are preserved by the hyperspace of compact subsets of a quasi-uniform space. In addition, we give a generalisation of the Morita-Zenor theorem by using compact symmetry. In studying the dual properties of quasi-metrizability, we shall investigate classes of both quasi-Nagata spaces and k-semi-stratifiable spaces. Basic properties and operations of these classes of topological spaces are discussed. Inter-relationships with other well-known classes of spaces such as semi-stratifiable spaces and stratifiable spaces are presented. We introduce a type of topological game called G(F)-game by using a filter F and, a sort of covering property, which is called game-compact, associated with this kind of game. We discover that each k-semi-stratifiable space is game.compact. More importantly, game-compactness can be applied to study the structure of upper semicontinuity of a multifunction. Consequently, the Choquet-Dolecki theorem is generalised. We also use Cp-theory to construct some examples of game-compact spaces which are not Dieudonné-complete. Meanwhile, a recent question in Cp-theory due to Arkhangel'skii is answered in the class of G-spaces defined by using game-theory Finally, we will apply quasi-uniform spaces to study the classical theory of function spaces. The idea of using generalised uniform spaces to study function spaces started in 1960s with Naimpally, Seyedin and Morales et al. However, they seem to have been unsuccessful: indeed, we make corrections to some of their main results. We shall consider fundamental questions such as: When is the family of continuous functions a closed subset of the function space? When is a function space "complete"? Two open questions posed by Papadopoulos in 1994 are answered negatively, and a general Ascoli theory is established by using compact symmetric and small-set symmetric quasi-uniform spaces. en
dc.language.iso en en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA9987065314002091 en
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 en
dc.title Asymmetric Topology and Topological Spaces Defined By Games en
dc.type Thesis en
thesis.degree.grantor The University of Auckland en
thesis.degree.level Doctoral en
thesis.degree.name PhD en
dc.rights.holder Copyright: The author en


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