Our subject matter consists of a survey of the major results concerning the topological space βN-N where N represents the space of natural numbers with the discrete topology, and βN the Stone-Čech compactification of N . We are mainly concerned with the results which were derived during the last ten years. When there is no advantage in restricting our work to the space N we work with an arbitrary discrete space X and finally formulate our results in terms of βN-N . In some cases, pre-1960 results concerning βN-N are obtained as special cases of the results we derive using an arbitrary discrete space X . The material presented is divided into four chapters. In Chapter I, we discuss certain subsets of βN-N which can be C*-embedded in other subsets of βN-N . This study leads to the conclusion that no proper dense subset of βN-N can be C*-embedded. In the second chapter we devise a general method of associating certain classes of points of βN-N with certain subalgebras of C(N) . The P-points of βN-N form one of these classes. The answer to R. S. Pierce's question, "Does there exist a point of βN-N which lies simultaneously in the closures of three pairwise disjoint open sets" is discussed in Chapter III. Finally in Chapter IV we present two proofs of the non-homogeneity of βN-N , without the use of the Continuum Hypothesis. / Science, Faculty of / Mathematics, Department of / Graduate

A hierarchy of separation axioms can be obtained by considering which axiom implies another. This thesis studies the properties of some separation axioms between T₀ and T₁ and investigates where each of the axioms belongs in this hierarchy. The behaviours of the axioms under strengthenings of topologies and cartesian products are considered. Given a set X, the family of all topologies defined on X is a complete lattice. A study of topologies which are minimal in this lattice with respect to a certain separation axiom is made. We consider certain such minimal spaces, obtain some characterizations and study some of their properties. / Science, Faculty of / Mathematics, Department of / Graduate

Let (X , ਹ) be a topological space. If < is a total ordering on X , then (X , ਹ, <) is said to be an ordered topological space if a subbasis for ਹ is the collection of all sets of the form {x ∊ x | x < t} or [x ∊ x | t < x} where t ∊ X . The pair (X , ਹ) is said to be an orderable topological space if there exists a total ordering, < , on X such that (X , ਹ, <) is an ordered topological space. Definition: Let T be a subspace of the real line ǀR . Let Q be the union of all non-trivial components of T , both of whose end points belong to C1ıʀ(C1ıʀ(T) -T). The following characterization of orderable sub-spaces of ǀR is due to M. E. Rudin. Theorem: Let T be a subspace of ǀR with the relativized usual topology. Then T is orderable if and only if T satisfies the following two conditions: (1) If T - Q is compact and (T-Q) ก Clıʀ(Q) = Ø then either Q = Ø or T - Q = Ø (2) If I is an open interval of ıʀ and p is an end point of I and if {p} U(I ก(T-Q)) is compact and {p} =Clıʀ(IกQ)ก C1ıʀ(I ก(T-Q)), then p ∉ T or {p} is a component of T. This theorem enables us to prove a conjecture of I.L. Lynn, namely Corollary: if T contains no open compact sets then T is totally orderable. If T is a subspace of an arbitrary ordered topological space a generalization of the theorem can be made. The generalized theorem is stated and some examples are given. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we prove a topological lemma on real valued random variables which implies the basic ingredients for the proof of the Fundamental Theorem of Asset Pricing in the two period case. In particular, previous results of Stricker and of Schachermayer are special cases of our result. Our proof is considerably shorter and more transparent than previous proofs of related special cases. / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory.

Thesis advisor: Kenneth S. Burch / Since their first experimental realizations in the 2000s, bulk electronic topological materials have been one of the most actively studied areas of condensed matter physics. Among the more recently discovered classes of topological materials are the Weyl semimetals whose low energy excitations behave like massless, relativistic particles with well-defined chirality. These material systems display exotic behavior such as surface Fermi arc states, and the chiral anomaly in which parallel magnetic and electric fields lead to an imbalance of left- and right-handed particles. Much of the research into these materials has focused on the electronic properties, but relatively little has been directed towards understanding the vibrational properties of these systems, or of the interplay between the electronic and vibrational degrees of freedom. Further, the technological potential of these materials is still underdeveloped, with the search for physical properties enhanced by the topological nature of these materials being sought after. In this dissertation we address both of these issues. In Chapters III and IV we present temperature dependent Raman investigations of the the Weyl semimetals WP2, NbAs, and TaAs. Measurements of the optical phonon linewidths are used to identify the available phonon decay paths, with ab-initio calculations and group theory used to aid the interpretation of these results. We find that some phonons display linewidths indicative of dominant decay into electron-hole pairs near the Fermi surface, rather than decay into acoustic phonons. In light of these results we discuss the role of phonon-electron coupling in the transport properties of these Weyl semimetals. In Chapter V, we discuss the construction of our "PVIC" setup for the measurement of nonlinear photocurrents. We discuss the experimental capabilities that the system was designed to possess, the operating principles behind key components of the system, and give examples of the operating procedures for using the setup. The penultimate chapter, Chapter VI, presents the results of photocurrent measurements using this setup on the Weyl semimetal TaAs. Through careful analysis of the photocurrent polarization dependence, we identify a colossal bulk photovoltaic effect in this material which exceeds the response displayed by previously studied materials by an order of magnitude. Calculations of the second-order optical conductivity tensor show that this result is consistent with the divergent Berry connection of the Weyl nodes in TaAs. In addition to these topics, Chapter II addresses the results of Raman measurements on thin film heterostructures of the topological insulator Bi2Se3 and the magnetic semiconductor EuS. By investigating the paramagnetic Raman signal in films with different compositions of EuS and Bi2Se3 we provide indirect evidence of charge transfer between the two layers. We also track the evolution of phonon energies with varying film thicknesses on multiple substrates which provides insight into the interfacial strain between layers. We conclude the dissertation in Chapter VII with a summary of the main results from each preceding chapter, and give suggestions for future experiments that further investigate these topics. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

This thesis investigates some of the properties of certain transformations. Some properties are considered in general; others, only in the xy-plane.

The purpose of this thesis is to investigate the relation Z for point sets in a topological space. There were two original goals which caused the study.

No description available.

Abstract Not Provided. / Thesis / Master of Science (MSc)