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THE STONE-^ECH COMPACTIFICATION by Russell C. Walker Department of Mathematics Carnegie-Mellon University- August, 1972 The Stone-(5ech Compactification by Russell C. Walker Carnegie-Mellon University ABSTRACT The Stone-fiech compactification PX has been a topic of increasing study since its introduction in 1937. The algebraic content of this research is collected in the 1960 textbook, Rings of continuous Functions, by L. Gillman and M. Jerison. Here we take a more purely topological viewpoint of the Stone-Cech compactification and attempt to collect the most important results which have emerged since Rings of Continuous Functions. The construction of pX is described in an historical perspective. The theory of Boolean algebras is developed and used as a tool, primarily in a detailed investigation of 0 IN and p 3N\IN. The relationships between a space X and its "growth" PX\X are examined, including the non-homogeneity of $X\X, the cellularity of pX\X, and mappings of pX to PX\X. The Glicksberg product theorem which characterizes the products such that 0(x X cc) = x (PX^cc ) and related results are presented. Finally, the Stone-£ech compactification is studied in a categorical context. INTRODUCTION This manuscript had its origins in the author*s fascination with L. Gillman and M. Jerison*s text, Rings of Continuous Functions, particularly with the Stone-Cech compactification. The number of research papers relating to the Stone-fiech compactification since the publication of Rings of Continuous Functions in 1960 shows that I am not alone in my interest. My main objective in this manuscript is to collect these results in a single source in order to make them more accessible to the mathematical community. However, the point of view here is more topological than that of Gillman and Jerison and little of the algebraic interest in the Stone-Cech compactification is apparent beyond the first chapter. Only a very few of the results here are new. Theorem 10.46 concerning epi-reflective hulls and the results in Sections 10.28-31 concerning pushouts in the category of completely regular spaces were first presented by the author to a categorical topology seminar at Carnegie-Mellon University in 1971. The proof of Theorem 10.22 characterizing the realcompact spaces as closed subspaces of products of real lines appears to be new. Finally, Lemma 2.26 is a Boolean algebra result which is crucial to the investigation of p W\W. The lemma first appeared in I. I. ParoviCenko*s 1963 paper as a generalization of a result of W. Rudin. However, Parovitfenko*s proof contains a gap. To i the best of my knowledge, the proof given here is the first | - complete proof of this result. ; This manuscript is intended to be both a reference for research mathematicians and to be a text for second level graduate students in topology. An extensive bibliography has been included as an aid to furthejr reading and the original i papers are cited in the text by year |and author. A list of exercises follows most chapters to supplement the text and to give the reader an opportunity to become familiar with the concepts. i Chapters 2, and 10 are introductory in nature. Chapter 1 contains the basic material about th^ Stone-Cech compactification and parts of the chapter will be repetitive for the reader already familiar with the concept. Chapter 2j is an introduction to Boolean algebras. Chapter 10 provides a brief introduction to category theory and discusses the Stcjne-Cech compactification in the categorical context of reflections. Each of Chapters 3 - 9 i develops a particular topic in the tjheory of Stone-fiech compactifications. Chapters 3 and 7 |are devoted to $ IN and its growth pM\3N. For the reader mainly interested in these two spaces, Corollary 4.30 which showis the existence of P-points in p ]N\IN and the construction used in the proof of Corollary i 6.16 comprise the material needed bettween Chapters 3 and 7. The major pattern of dependence among the chapters is given | • • below. ii Chapter 1 Chapter 8 Chapter 9 Chapter 2 Chapter 10 V Chapter 3 Corollaries 4.30,6.16 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Because the later chapters are essentially independent, the manuscript can be used as a text in several ways. As a one semester second level graduate course, portions of Chapters 1, 2, 3, and 10 together with one of the more specialized chapters would appear to be a good combination of about the right length. The reader already familiar with Rings of Continuous Functions will need only to read Chapter 2 and part of Chapter 3 in order to have easy access to any of the later chapters. As a two semester course, the student who has completed an introductory topology course should be able to cover the complete manuscript. iii ACKNOWLEDGEMENTS I would like to express my gratitude to Professor Stanley P. Franklin under whose guidance this dissertation has been written. Without his advice and encouragement, the project would have been nearly impossible as well as much less enjoyable. I would also like to thank Greg Naber and Barbara Thomas for their constructive comments and criticism. The author would also like to express his gratitude to Carnegie-Mellon University for financial support during the writing of this manuscript and also to Professors Richard A. Moore and Maury Weir for their encouragement during the first years of graduate study. Finally, I would like to thank Nancy Colmer for her excellent typing of the final manuscript. iv TABLE OF CONTENTS 1. Development of the Stone-£ech Compactification 1 2. Boolean Algebras 76 3. On pJN and IN* 131 4. Non-homogeneity of Growths 179 5. Cellularity of Growths 218 6. Mappings of pX to X* 245 7. p IN Revisited 281 8« Product Theorems 293 9. Some Connectedness Results 347 10. pX in Categorical Perspective 361 Bibliography 445 Author Index 470 List of Symbols 472 Index 474 :-•••• CHAPTER ONE: DEVELOPMENT OF THE STONE-^ECH COMPACTIFICATION 1.1. A compactification of a topological space X is a compact space K together with an embedding e : X-—with e [X] dense in K. We will usually identify X with e [X] and consider X as a subspace of K. Our main topic is a very special type of compactification -- one in which X is embedded in such a way that every bounded, real-valued continuous function on X will extend continuously to the compactification. Such a compactification of X will be called a Stone-Cech compacti- fication and will be denoted by j8x. In this chapter, several constructions of #X will be examined. We will find that j8x is a useful device to study relationships between topological characteristics of X and the algebraic structure of the real- valued continuous functions defined on X and that many topological properties of X can be translated into properties of $X. The discussion will proceed in a rough approximation of historical order, although many anachronisms have been included to smooth the way and at the same time to introduce material that will be useful later. Much of the first chapter will be a review of material developed in the classic L. Gillman and M. Jerison text, Rings of Continuous Functions which will henceforth be referred to by [GJ]. The development will be almost entirely self-contained. Only a very few results which rely on the more algebraic approach 2 of [GJ] will be stated without proof. The basic reference for results in general topology will be J. Dugundji*s text, Topology, which will be referred to by [D]. 1.2. For the most part, notation will be as in [GJ]. The ring of all real-valued continuous functions defined on a space X will be denoted by C(X) and the subring consisting of all bounded members of C(X) will be denoted by C*(X). We will say that a subspace S of X is C-embedded (resp. C*-embedded) in X if every member f of C(S) (resp. C*(S)) extends to a member g of C(X) (resp. C (X)). The following diagram illustrates C-embedding: e The symbol # indicates that the composition of g with the embedding e is equal to the mapping f. In such an instance, we will say that the diagram is commutative. The set of points of X where a member f of C(X) is equal to zero is called the zero-set of f and will be denoted by Z(f). We will frequently say that f vanishes at x to mean f(x) =0. The complement of the zero-set Z(f) is called a cozero-set and is denoted by Cz(f). The collections of all zero-sets and all cozero-sets of X will be denoted by 3 Z[X] and CZ[X], respectively. The ring and lattice operations in the rings C(X) and C (X) will be defined pointwise and it will be convenient to be familiar with such relationships as Z(|f| + |g|) = Z(f) n z(g) and {xeX : f(x) 2 1} = Z((f-1) A 0). Note that in the second equation, the symbol "1" is used to represent the function which is constantly equal to 1. The symbol f"*1 will be reserved to denote the reciprocal of a member f of C(X) if the reciprocal is defined, i.e. if f does not vanish anywhere on X. The term mapping will always refer to a continuous function. The composition of two functions f and g will be denoted by go f and inverse images under a function f will be written f (S) .
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