Rings of Continuous Functions: 43 (The University Series in Higher Mathematics) - Hardcover

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 dis­ 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 con­ 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.

The objective of this book is the systematic study of the ring of all real valued continuous functions on arbitrary topological spaces. Great emphasis is placed on the study of ideals, and on the associated residue class rings. Questions of extending continuous functions from a subspace to the entire space arise as a necessary adjunct and are dealt with in considerable detail. Many problems provide additional information about the material covered in the text.