This book is addressed to those who know the meaning of each word in the title: none is defined in the text. The reader can estimate the knowledge required by looking at Chapter 0; he should not be disยญ couraged, however, if he finds some of its material unfamiliar or the presentation rather hurried. Our objective is a systematic study of the ring C(X) of all real-valued continuous functions on an arbitrary topological space X. We are conยญ cerned with algebraic properties of C(X) and its subring C*(X) of bounded functions and with the interplay between these properties and the topology of the space X on which the functions are defined. Major emphasis is placed on the study of ideals, especially maximal ideals, and on their associated residue class rings. Problems of extending continuous functions from a subspace to the entire space arise as a necessary adjunct to this study and are dealt with in considerable detail. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5 and the beginning of Chapter 10, presents the fundamental aspects of the subject insofar as they can be discussed without introducing the Stone-Cech compactification. In Chapter 3, the study is reduced to the case of completely regular spaces
CONTENT
1 Functions on a Topological Space -- 2 Ideals and z-Filters -- 3 Completely Regular Spaces -- 4 Fixed Ideals. Compact Spaces -- 5 Ordered Residue Class Rings -- 6 The Stone-?ech Compactification -- 7 Characterization of Maximal Ideals -- 8 Realcompact Spaces -- 9 Cardinals of Closed Sets in ?X -- 10 Homomorphisms and Continuous Mappings -- 11 Embedding in Products of Real Lines -- 12 Discrete Spaces. Nonmeasurable Cardinals -- 13 Hyper-Real Residue Class Fields -- 14 Prime Ideals -- 15 Uniform Spaces -- 16 Dimension -- Notes -- List of Symbols
