This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.

Inverse systems, inverse limit spaces, and bonding maps are defined. An investigation of the properties that an inverse limit space inherits, depending on the conditions placed on the factor spaces and bonding maps is made. Conditions necessary to ensure that the inverse limit space is compact, connected, locally connected, and semi-locally connected are examined. A mapping from one inverse system to another is defined and the nature of the function between the respective inverse limits, induced by this mapping, is investigated. Certain restrictions guarantee that the induced function is continuous, onto, monotone, periodic, or open. It is also shown that any compact metric space is the continuous image of the cantor set. Finally, any compact Hausdorff space is characterized as the inverse limit of an inverse system of polyhedra.

by Chi-keung Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Introduction --- p.1 / Chapter Chapter 0. --- Preliminary --- p.4 / Chapter 0.1 --- Topological vector spaces / Chapter 0.2 --- Ordered vector spaces / Chapter 0.3 --- Ordered normed spaces / Chapter 0.4 --- Ordered topological vector spaces / Chapter 0.5 --- Ordered bornological vector spaces / Chapter Chapter 1. --- Results on Ordered Normed Spaces --- p.23 / Chapter 1.1 --- Results on e∞-spaces and e1-spaces / Chapter 1.2 --- Complemented subspaces of ordered normed spaces / Chapter 1.3 --- Half injections and Half surjections / Chapter 1.4 --- Strict quotients and strict subspaces / Chapter Chapter 2. --- Helley's Selection Theorem and Local Reflexivity Theorem of order type --- p.55 / Chapter 2.1 --- Helley's selection theorem of order type / Chapter 2.2 --- Local reflexivity theorem of order type / Chapter Chapter 3. --- Operator Modules and Ideal Cones --- p.68 / Chapter 3.1 --- Operator modules and ideal cones / Chapter 3.2 --- Space cones and space modules / Chapter 3.3 --- Injectivity and surjectivity / Chapter 3. 4 --- Dual and pre-dual / Chapter Chapter 4. --- Topologies and Bornologies Defined by Operator Modules and Ideal Cones --- p.95 / Chapter 4.1 --- Generalized polars / Chapter 4.2 --- Topologies and bornologies defined by β and ε / Chapter 4. 3 --- Injectivity and generating topologies / Chapter 4.4 --- Surjectivity and generating bornologies / Chapter 4.5 --- The solid property and the generating topologies / Chapter 4.6 --- The solid property and the generating bornologies / Chapter Chapter 5. --- Semi-norms and Bounded disks defined by Operator Modules and Ideal Cones --- p.129 / Chapter 5.1 --- Results on semi-norms / Chapter 5.2 --- Results on bounded disks / References --- p.146 / Notations --- p.149

Yung Hon-wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 133-139). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Density Theorems in Vector Optimization --- p.7 / Chapter 1.1 --- Preliminary --- p.7 / Chapter 1.2 --- The Arrow-Barankin-Blackwell Theorem in Normed Spaces --- p.14 / Chapter 1.3 --- The Arrow-Barankin-Blackwell Theorem in Topolog- ical Vector Spaces --- p.27 / Chapter 1.4 --- Density Results in Dual Space Setting --- p.32 / Chapter 2 --- Density Theorem for Super Efficiency --- p.45 / Chapter 2.1 --- Definition and Criteria for Super Efficiency --- p.45 / Chapter 2.2 --- Henig Proper Efficiency --- p.53 / Chapter 2.3 --- Density Theorem for Super Efficiency --- p.58 / Chapter 3 --- Connectedness Results in Vector Optimization --- p.63 / Chapter 3.1 --- Set-valued Maps --- p.64 / Chapter 3.2 --- The Contractibility of the Efficient Point Sets --- p.67 / Chapter 3.3 --- Connectedness Results in Vector Optimization Prob- lems --- p.83 / Chapter 4 --- Error Bounds In Normed Spaces --- p.90 / Chapter 4.1 --- Error Bounds of Lower Semicontinuous Functionsin Normed Spaces --- p.91 / Chapter 4.2 --- Error Bounds of Lower Semicontinuous Convex Func- tions in Reflexive Banach Spaces --- p.100 / Chapter 4.3 --- Error Bounds with Fractional Exponents --- p.105 / Chapter 4.4 --- An Application to Quadratic Functions --- p.114 / Bibliography --- p.133

In this dissertation separation and selection theorems are presented. It begins by presenting a detailed proof of the Inductive Definability Theorem of D. Cenzer and R.D. Mauldin, including their boundedness principle for monotone coanalytic operators.

The Buoyancy Theorem states that a compact set is buoyant if every point of the compact set has a neighborhood whose intersection with the compact set is buoyant. In this paper, the Buoyancy Theorem is used to prove several standard results involving compact sets. The proof of such a result may be a direct application of the Buoyancy Theorem or the proof may rely on a certain compactness argument which follows from the Buoyancy Theorem. The last application in this paper is such an example. The method used is to, first of all, define a buoyancy on the compact set; secondly, show that every point of the compact set has a neighborhood whose intersection with the compact set is buoyant; and finally, apply the Buoyancy Theorem to conclude that the compact set is buoyant.

Thesis (Ph. D.)--University of Rhode Island, 2005. / Typescript. Includes bibliographical references (leaves 87-89).

The aim of this thesis is to establish the principal properties for the theory of ordered compactifications relating to connectedness and to provide particular examples. The initial idea of this subject is based on the notion of the Stone-Cech compactification.The ordered Stone-Cech compactification oX of an ordered topological space X is constructed analogously to the Stone-Cech compactification X of a topological space X, and has similar properties. This technique requires a conceptual understanding of the Stone-Cech compactification and how its product applies to the construction of ordered topological spaces with continuous increasing functions. Chapter 1 introduces background information. Chapter 2 addresses connectedness and compactification. If (A;B) is a separation ofa topological space X, then (A 8 B) = A 8 B, but in the ordered setting, o(A 8 B)need not be oA 8 oB. We give an additional hypothesis on the separation (A;B) tomake o(A 8 B) = oA 8 oB. An open question in topology is when is X -X = X. Weanswer the analogous question for ordered compactifications of totally ordered spaces. So, we are concerned with the remainder, that is, the set of added points oX -X. Wedemonstrate the topological properties by using lters. Moreover, results of lattice theory turn out to be some of the basic tools in our original approach. In Chapter 3, specific examples and counterexamples are given to illustrate earlierresults.

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