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A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers

Description: Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.
Date: August 2017
Creator: Allen, Cristian Gerardo
Partner: UNT Libraries

Hyperspace Topologies

Description: In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.
Date: August 2001
Creator: Freeman, Jeannette Broad
Partner: UNT Libraries

Metric Half-Spaces

Description: This paper is a study of some of the basic properties of the metric half-space topology, a topology on a set which is derived from a metric on the set. In the first it is found that in a complete inner product space, the metric half-space topology is the same as one defined in terms of linear functionals on the space. In the second it is proven that in Rn the metric half-space topology is the same as the usual metric topology. In the third theorem it is shown that in a certain sense the nature of the metric halfspace topology generated by a norm on the space determines whether the norm is quadratic, that is to say, whether or not there exists an inner product on the space with the property that |x|^2=(x,x) for all x in the space.
Date: May 1972
Creator: Dooley, Willis L.
Partner: UNT Libraries

Topologies on Complete Lattices

Description: One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.
Date: December 1973
Creator: Dwyer, William Karl
Partner: UNT Libraries

Spaces of Closed Subsets of a Topological Space

Description: 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.
Date: August 1974
Creator: Leslie, Patricia J.
Partner: UNT Libraries

Synchronization in complex networks

Description: Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.
Date: December 12, 2007
Creator: Arenas, A.; Diaz-Guilera, A.; Moreno, Y.; Zhou, C. & Kurths, J.
Partner: UNT Libraries Government Documents Department

The Computation of Ultrapowers by Supercompactness Measures

Description: The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.
Date: August 1999
Creator: Smith, John C.
Partner: UNT Libraries

Algebraic Numbers and Topologically Equivalent Measures

Description: A set-theoretical point of view to study algebraic numbers has been introduced. We extend a result of Navarro-Bermudez concerning shift invariant measures in the Cantor space which are topologically equivalent to shift invariant measures which correspond to some algebraic integers. It is known that any transcendental numbers and rational numbers in the unit interval are not binomial. We proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. Algebraic integers of degree 2 are proved not to be binomial numbers. A few compositive relations having to do with algebraic numbers on the unit interval have been studied; for instance, rationally related, integrally related, binomially related, B1-related relations. A formula between binomial numbers and binomial coefficients has been stated. A generalized algebraic equation related to topologically equivalent measures has also been stated.
Date: December 1983
Creator: Huang, Kuoduo
Partner: UNT Libraries

Dimension Theory

Description: This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.
Date: August 1986
Creator: Frere, Scot M. (Scot Martin)
Partner: UNT Libraries

Hyperspaces

Description: 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.
Date: December 1976
Creator: Voas, Charles H.
Partner: UNT Libraries

A Topological Uniqueness Result for the Special Linear Groups

Description: The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.
Date: August 1997
Creator: Opalecky, Robert Vincent
Partner: UNT Libraries

Topological Landscapes: A Terrain Metaphor for ScientificData

Description: Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context, the use of metaphors plays an important role, since they make complex information easily accessible by using commonly known concepts. In this paper we propose a new metaphor, called 'Topological Landscapes', which facilitates understanding the topological structure of scalar functions. The basic idea is to construct a terrain with the same topology as a given dataset and to display the terrain as an easily understood representation of the actual input data. In this projection from an n-dimensional scalar function to a two-dimensional (2D) model we preserve function values of critical points, the persistence (function span) of topological features, and one possible additional metric property (in our examples volume). By displaying this topologically equivalent landscape together with the original data we harness the natural human proficiency in understanding terrain topography and make complex topological information easily accessible.
Date: August 1, 2007
Creator: Weber, Gunther H.; Bremer, Peer-Timo & Pascucci, Valerio
Partner: UNT Libraries Government Documents Department

Topological Properties of Chains

Description: The purpose of this paper is to define and investigate some of the properties of chains. Particular attention is given to a natural topology for chains, called the interval topology, and how the chain properties and topological properties of chains affect each other.
Date: January 1966
Creator: Womack, Robert A.
Partner: UNT Libraries

Topological Groups

Description: The notion of a topological group follows naturally from a combination of the properties of a group and a topological space. Since a group consists of a set G of elements which may be either finite or infinite and since this is also common to a topological space, a question is opened as to whether or not it is possible to assign a topology to a set of elements which form a group under a certain operation. Now it is possible to assign a topology to any set of elements if no restriction is placed on the topology assigned and hence this study would be of little value from the standpoint of the group itself. If however it is required that the group operation be continuous in the topological space then a very interesting theory is developed.
Date: May 1960
Creator: Carry, Laroy Ray
Partner: UNT Libraries

Semitopological Groups

Description: This thesis is a study of semitopological groups, a similar but weaker notion than that of topological groups. It is shown that all topological groups are semitopological groups but that the converse is not true. This thesis investigates some of the conditions under which semitopological groups are, in fact, topological groups. It is assumed that the reader is familiar with basic group theory and topology.
Date: December 1971
Creator: Scroggs, Jack David
Partner: UNT Libraries

Separation Properties

Description: The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.
Date: December 1970
Creator: Garvin, Billy Ray
Partner: UNT Libraries

Relative Extreme Points

Description: In this paper, elementary properties of relative extreme points are investigated. The properties are defined in linear and topological terms. Proofs of many of these properties require the use of topological concepts.
Date: January 1966
Creator: Matthews, William J.
Partner: UNT Libraries