Introduction to Topology

Second Edition

Theodore W. Gamelin, Robert Everist Greene

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486320182
• Edition: 2
• Topic: Mathematics
• Subtopic: Topology
• Index: Mathematics

One of the most important milestones in mathematics in the twentieth century was the development of topology as an independent field of study and the subsequent systematic application of topological ideas to other fields of mathematics.
While there are many other works on introductory topology, this volume employs a methodology somewhat different from other texts. Metric space and point-set topology material is treated in the first two chapters; algebraic topological material in the remaining two. The authors lead readers through a number of nontrivial applications of metric space topology to analysis, clearly establishing the relevance of topology to analysis. Second, the treatment of topics from elementary algebraic topology concentrates on results with concrete geometric meaning and presents relatively little algebraic formalism; at the same time, this treatment provides proof of some highly nontrivial results. By presenting homotopy theory without considering homology theory, important applications become immediately evident without the necessity of a large formal program.
Prerequisites are familiarity with real numbers and some basic set theory. Carefully chosen exercises are integrated into the text (the authors have provided solutions to selected exercises for the Dover edition), while a list of notations and bibliographical references appear at the end of the book.

Metric Spaces

ONE

The ideas of “metric” and “metric space,” which are the subject matter of this chapter, are abstractions of the concept of distance in Euclidean space. These abstractions have turned out to be particularly fundamental and useful in modern mathematics; in fact, the aspects of the Euclidean idea of distance retained in the abstract version are precisely those that are most useful in a wide range of mathematical activities. The determination of this usefulness was historically a matter of experience and experiment. By now, the reader can be assured, the mathematical utility of the metric-space information developed in this chapter entirely justifies its careful study.

Sections 1 through 6 of this chapter are devoted to the basic definitions and main theorems about metric spaces in general. Among the theorems established, two are especially substantial: the result called the Baire Category Theorem, in Section 2, and the equivalence of compactness and sequential compactness, in Section 5. The material in these first six sections is basic to modem analysis.

Sections 7 through 9 treat more specialized topics. Section 7 introduces some special classes of metric spaces—the normed linear spaces and Banach spaces—that are particularly important in applications. These spaces have not only th...