The Stone-Čech Compactification

ISBN-13: 9783642619373 / Angielski / Miękka / 2011 / 334 str.

R.C. Walker

Opis

Recent research has produced a large number of results concerning the Stone-Cech compactification or involving it in a central manner. The goal of this volume is to make many of these results easily accessible by collecting them in a single source together with the necessary introductory material. The author's interest in this area had its origin in his fascination with the classic text Rings of Continuous Functions by Leonard Gillman and Meyer Jerison. This excellent synthesis of algebra and topology appeared in 1960 and did much to draw attention to the Stone-Cech compactification {3X as a tool to investigate the relationships between a space X and the rings C(X) and C*(X) of real-valued continuous functions. Although in the approach taken here {3X is viewed as the object of study rather than as a tool, the influence of Rings of Continuous Functions is clearly evident. Three introductory chapters make the book essentially self-contained and the exposition suitable for the student who has completed a first course in topology at the graduate level. The development of the Stone Cech compactification and the more specialized topological prerequisites are presented in the first chapter. The necessary material on Boolean algebras, including the Stone Representation Theorem, is developed in Chapter 2. A very basic introduction to category theory is presented in the beginning of Chapter 10 and the remainder of the chapter is an introduction to the methods of categorical topology as it relates to the Stone-Cech compactification."

Spis treści

1. Development of the Stone-?ech Compactification.- Completely Regular Spaces.- ?X and the Extension of Mappings.- ?-Filters and ?-Ultrafilters.- ?X and Maximal Ideal Spaces.- Spaces of ?-Ultrafilters.- Characterizations of ?X.- Generalizations of Compactness.- F-Spaces and P-Spaces.- Other Approaches to ?X.- Exercises.- 2. Boolean Algebras.- The Stone Representation Theorem.- Two Examples.- The Completion of a Boolean Algebra.- Separability in Boolean Algebras.- Exercises.- 3. On ?? and ?*.- The Cardinality of ??.- The Clopen Sets of ?? and ?*.- A Characterization of ?*.- Types of Ultrafilters and the Non-Homogeneity of ?*.- Exercises.- 4. Non-Homogeneity of Growths.- Types of Points in X*.- C-Points and C*-Points in X*.- P-Points in X*.- Remote Points in X*.- The Example of ??.- Exercises.- 5. Cellularity of Growths.- Lower Bounds for the Cellularity of X*.- n-Points and Uniform Ultrafilters.- n-Points and Compactifications of ?.- Exercises.- 6. Mappings of ?X to X*.- C*-Embedding of Images.- Retractive Spaces.- Growths of Compactifications.- Mappings of ?D and other Extremally Disconnected Spaces.- Exercises.- 7. ?? Revisited.- ?*\ is not Normal.- An Example Concerning the Banach-Stone Theorem.- A Point of ?* with c Relative Types.- Types, ?*-Types, and P-Points.- Minimal Types and Points with Finitely Many Relative Types.- Exercises.- 8. Product Theorems.- Glicksberg’s Theorem for Finite Products.- The Product Theorem for Infinite Products.- Assorted Product Theorems.- The ?-Analogue: An Open Question.- Exercises.- 9. Local Connectedness, Continua, and X*.- Compactifications of Locally Connected Spaces.- A Non-Metric Indecomposable Continuum.- Continua as Growths.- Exercises.- 10. ?X in Categorical Perspective.- Categories and Functors.- Reflective Subcategories of the Category of Hausdorff Spaces.- Adjunctions in Reflective Subcategories.- Perfect Mappings.- Projectives.- Exercises.- Author Index.- List of Symbols.