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

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.

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The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets.

A Dissertation Submitted of the Faculty of Science, University of the Witwatersrand, Johannesburg in Partial Fulfilment of the Requirements for the Degree of Master of Science / Let 4 be the class of pairs of loc ..My onvex spaces (X,V) “h ‘ch are such that every closed graph linear ,pp, 1 from X into V is continuous. It B is any class of locally . ivex l.ausdortf spaces. let & w . (X . (X.Y) e 4 for ,11 Y E B). " ‘his expository dissertation, * (B) is investigated, firstly i r arbitrary B . secondly when B is the class of C,-complete paces and thirdly whon B is a class of locally convex webbed s- .ces

by Ma Mang Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1989. / Bibliography: leaf 77.

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