返回Google圖書搜尋

Asymmetric Topology and Topological Spaces Defined by Games

出版	University of Auckland, 1999
URL	http://books.google.com.hk/books?id=qfdUMQAACAAJ&hl=&source=gbs_api

註釋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.