Rings of Continuous Functions PDF

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.