Open Collections

Abstract

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.