The purpose of this paper is to examine selected topologies, the Vietoris topology in particular, on S(X), the collection of nonempty, closed subsets of a topological space X. Characteristics of open and closed subsets of S(X), with the Vietoris topology, are noted. The relationships between the space X and the space S(X), with the Vietoris topology, concerning the properties of countability, compactness, and connectedness and the separation properties are investigated. Additional topologies are defined on S(X), and each is compared to the Vietoris topology on S(X). Finally, topological convergence of nets of subsets of X is considered. It is found that topological convergence induces a topology on S(X), and that this topology is the Vietoris topology on S(X) when X is a compact, Hausdorff space.

Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}

We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.

This thesis concerns two-point sets, which are subsets of the real plane which intersect every line in exactly two points. The existence of two-point sets was first shown in 1914 by Mazukiewicz, and since this time, the properties of these objects have been of great intrigue to mathematicians working in both topology and set theory. Arguably, the most famous problem about two-point sets is concerned with their so-called "descriptive complexity"; it remains open, and it appears to be deep. An informal interpretation of the problem, which traces back at least to Erdos, is: The term "two-point" set can be defined in a way that it is easily understood by someone with only a limited amount of mathemat- ical training. Even so, how hard is it to construct a two-point set? Can one give an effective algorithm which describes precisely how to do so? More formally, Erdos wanted to know if there exists a two-point set which is a Borel subset of the plane. An essential tool in showing the existence of a two-point set is the Axiom of Choice, an axiom which is taken to be one of the basic truths of mathematics.

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by He Xinggang. / "December 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 69-73). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Li Jian Lin. / "November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 85-90) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Lai, Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 83-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Spectral sets in Rd --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Fundamental domains and convex sets in Rd --- p.15 / Chapter 2.3 --- Finite union of cubes --- p.20 / Chapter 3 --- Spectral theory on discrete groups --- p.27 / Chapter 3.1 --- Finite groups and Zd --- p.28 / Chapter 3.2 --- Rational spectrums and tiling sets --- p.32 / Chapter 3.3 --- Fuglede´ةs Problem in R1 --- p.37 / Chapter 3.4 --- "Failure of Fuglede´ةs Conjecture in Rd, d >3" --- p.42 / Chapter 4 --- Self-similar tiles in R1 --- p.49 / Chapter 4.1 --- Basics of self-similar tiles --- p.49 / Chapter 4.2 --- Self-similar tile digit sets and spectral problem --- p.52 / Chapter 4.3 --- Kenyon criterion --- p.55 / Chapter 5 --- Spectral self-similar measures --- p.66 / Chapter 5.1 --- Spectral self-similar measures --- p.66 / Chapter 5.2 --- One-dimensional self-similar measures --- p.72 / Chapter 5.3 --- General properties of spectral measures --- p.80 / Bibliography --- p.83

Wong, Wing Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 66-67). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background Study --- p.4 / Chapter 2.1 --- Basic Definitions and Notations --- p.5 / Chapter 2.2 --- Tarjan and Trojanowski's algorithm --- p.6 / Chapter 2.2.1 --- Techniques --- p.6 / Chapter 2.2.2 --- Algorithm --- p.8 / Chapter 2.3 --- "Fomin, Grandoni and Kratsch's Algorithm" --- p.9 / Chapter 2.3.1 --- Techniques --- p.9 / Chapter 2.3.2 --- Algorithm --- p.14 / Chapter 3 --- Improvements --- p.18 / Chapter 3.1 --- Tarjan and Trojanowski´ةs Algorithm --- p.18 / Chapter 3.1.1 --- Correctness and Running Time Analysis --- p.28 / Chapter 3.1.2 --- Improvement --- p.30 / Chapter 3.1.3 --- Using more weights --- p.35 / Chapter 3.2 --- The First Algorithm --- p.37 / Chapter 3.2.1 --- Standard Analysis --- p.37 / Chapter 3.2.2 --- Measure and Conquer --- p.38 / Chapter 3.2.3 --- Using more weights --- p.42 / Chapter 3.3 --- The Second Algorithm --- p.43 / Chapter 3.3.1 --- Running Time Analysis --- p.44 / Chapter 3.3.2 --- Using More Weights --- p.45 / Chapter 3.4 --- The Third Algorithm --- p.46 / Chapter 4 --- Lower Bounds --- p.52 / Chapter 4.1 --- Tarjan and Trojanowski's Algorithm --- p.52 / Chapter 4.2 --- The First Algorithm --- p.55 / Chapter 4.3 --- The Second Algorithm --- p.58 / Chapter 4.4 --- The Third Algorithm --- p.60 / Chapter 5 --- Conclusion --- p.63 / Bibliography --- p.66

This study investigated the distribution of bibliographic records amongst online bibliographic databases. The topic of Fuzzy Set Theory over the period of 1965 to 1993 was chosen to provide the case study for this investigation. From the DIALOG database host, searches were conducted on 114 databases to determine the number of journal article records relating to the topic of Fuzzy Sets. Both the number of records in each database, as well as the overlap of coverage between the databases were calculated. Six counting techniques were developed to allocate records to databases based on different methods for handling records that were duplicated between databases. When duplicate records are included, the top database accounts for 19% of the records; when duplicates are removed, the top database was found to account for 37% of the records. The distribution of records in databases was found to conform to the Bradford-Zipf hyperbolic distribution. Various other analyses were undertaken including: the duplicate records themselves, the total size of the DIALOG database system over time and the density of Fuzzy Set records in databases over time. A secondary aim of this study was to perform an informetric study on the literature of Fuzzy Set Theory itself. Results obtained include an analysis of the growth of the Fuzzy Set literature, an analysis of the journals covering the topic of Fuzzy Sets, an analysis of the terminology used in describing topics related to Fuzzy Sets. Also, the Ulrich's database was used to provide a subject classification of the journals to analyse the diffusion of the topic of Fuzzy Sets into other disciplines. Apart from the discipline of mathematics, the top disciplines into which Fuzzy Sets have diffused were found to be applied physics, systems and computing. The third aim of the thesis was to refine and develop the methodology used to perform large scale informetric studies using data from a variety of online bibliographic databases. Commercially available software was used wherever possible, but where this was not possible or infeasible, custom written programs were developed to perform various steps in the methodology.