![]() ![]() The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. Hypercompactness is studied in relation to compactness and collective compactness of sets of mappings.Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The properties of the hypergraph are related to continuity and uniform continuity, giving rise to theorems similar to closed graph theorems. The last two chapters deal with properties of mappings and sets of mappings, introducing two new concepts - the hypergraph of a mapping and a hypercompact set of mappings. The question of when two uniform structures induce the same topology on sets of subsets of particular kinds is investigated, and finally a new condition is provided for a uniform structure to be unique in the sense that no other uniform structure on the given set can induce the same topology on the set of subsets. A survey of known results is given and the relationships between them are discussed. ![]() Another subject which has attracted some interest recently is the comparability of the topologies induced on the set of subsets by various uniform structures on the original set. Wendy Robertson, and the analogous fundamental family condition, which bears a relationship to hypercompleteness similar to that of the filter condition to completeness. A study is made of two conditions on a mapping between uniform spaces, the filter condition, introduced by Dr. Kelley's notion of a fundamental family of subsets is applied to uniform spaces and the Hausdorff completion of the hyperspace is constructed by means of fundamental families. Making a completely different approach, J. Certain Hausdorff completions are shown to be uniformly embedded in the 'hyperhyperspace', and some generalisations are proved of results of the Robertsons on sets of compact subsets. Here a study is made of hyperspaces of Hausdorff completions and Hausdorff completions of hyperspaces, and, in particular, of a case where these spaces derive from two related uniform structures on a set. Historically the hyperspace has probably derived its importance mainly from the concept of hypercompleteness. Sometimes the results are for more specialised spaces - metric spaces, normed spaces or topological vector spaces, or for the more general proximity spaces. This thesis is concerned with the properties and uses of the so-called Hausdorff uniform structure on the set of subsets of a uniform space. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |