Construction of topological spaces via resolutions
Abstract (Summary)Restricted Item. Print thesis available in the University of Auckland Library or available through Inter-Library Loan. Resolutions are a topological construction method that evolved out of the Russian school of topologists during the 1960's and 1970's; they modify a topological (or direct) sum of spaces to allow for interaction between the topologies of the constituent spaces of the sum. As such, if the index set of the sum is a topological space, the behaviour of resolutions varies on a point-by-point basis as each point of the index set is studied. This conceptualisation of resolutions has been expressed by Watson, Richardson et al., as a 'magnification' process similar to that of studying a space under a magnifying glass. At first glance, the space that is the index set - call it the base space - carries only its own topological properties. However, as each point of the base space is examined, it may reveal a 'greater structure'; namely, under the scrutiny of a magnifying glass, the presence of a whole new space may be revealed where before there was thought to be only a single point of the base space. Carrying on such a scrutiny to every point of the base space, some points may reveal a greater structure, while others may remain a singleton even under the power of the magnifying glass. Having found at which points of the base space a 'greater structure' lies, the question of interest becomes "how do the greater structures revealed on a given base space interact with each other, and with those parts of the base space that have no greater structure?" The answer to this question is the study of the class of topological spaces known as resolutions. The 'greater structures' are termed fibres of the resolution, which is the topological sum of these structures, or fibres. Of course, those points of the base space that had no 'greater structure' are deemed to trivially be fibres themselves, so that every point of the original index set (or base space) has some structure in the sum (or resolution). The glue that holds the 'greater structures' together is a family of multifunctions (or set-valued functions), with one multifunction mapping all but one point of the base space into a fixed fibre. Analogously resolutions could be pictured as a furry animal, say a cat. From a distance, the surface of the cat appears smooth; however, upon closer scrutiny it is seen to be made up of millions of tiny hairs. Those parts of the skin visible between individual hairs represent the points of the skin (the base space) that had no greater structure (the hair, or fibre). This thesis, after a brief introduction to the special resolutions of Fedor?uk and Watson, investigates the definition and construction of a form of general resolution. Comparison is made with Ul'janov's general resolutions, which are a generalisation of Fedor?uk's special resolutions given in 1975 to permit product space structures within a resolution. This occupies chapters 1 and 2. The general theme of the thesis is then entered, being a systematic effort to give necessary and sufficient conditions for various topological properties to hold in a resolution, in relation to similar properties holding in the base space and fibres and other properties on the multifunctions associated with each fibre. Chapter 3 examines the internal structures of a resolution, especially the projection maps. A characterisation of when the map ?, which projects the resolution onto its base space, is a closed map is given. Notions of boundary and interior of the fibre, and hence resolution, are defined and explored briefly; both these and ? being a closed map are used heavily throughout the remainder of the thesis. Examination of the behaviour of topological properties in resolutions is then split into those properties that apply 'globally' to the entire space, in Chapter 4, and those that apply 'locally' in neighbourhoods of a point, in chapter 5. Sufficient conditions for normality and perfect normality, and generalised forms of metacompactness, paracompactness, and compactness are the primary results of the former chapter. The application of these results to the construction of a Dowker resolution is then explored in Section 4.5; this is representative of the possibility of application of resolutions to other known topological problems. A variety of examples of non-metrisable manifolds, culminating in the reproduction of a family of such manifolds by Gartside, Good, Knight and Mohamad and one such by M. E. Rudin, are the highlight of the latter chapter.
School Location:New Zealand
Source Type:Master's Thesis
Date of Publication:01/01/2004