The authors of *First Concepts of Topology* demonstrate the power, the flavor and the adaptability of topology, one of the youngest branches of mathematics, in proving so-called existence theorems. An existence theorem asserts that a solution to some given problem exists; thus it assures those who hunt for a solution that their labors may not be in vain. Since existence theorems are frequently basic to the structure of a mathematical subject, the applications of topology to the proofs of these theorems constitute a unifying force for large areas of mathematics.

In Part I of this monograph an existence theorem governing a large class of one-dimensional problems is treated; all the important ingredients in its proof, such as continuity of functions, compactness and connectedness of point sets, are developed an illustrated. In Part II, its two-dimensional analogue is carefully built via the necessary generalizations of the one-dimensional tools and concepts. The results are applied to such fundamental mathematical objects as zeros of polynomials, fixed points of mappings, and singularities of vector fields.

### Table of Contents

Introduction

Part I: Existence Theorems in Dimension 1

Part II. Existence Theorems in Dimension 2

Solutions for Exercises

Index

### About the Authors

**William G. Chinn** graduated from the University of California at Berkeley in 1941. After military service in World War II as a pharmaceutical technician and then as a weather forecaster, he resumed graduate study.

Mr. Chinn taught both in junior and senior high schools in San Francisco for 15 years. He maintained a mathematics “laboratory” for the gifted and was for three years assistant to the Curriculum Coordinator of the San Francisco Unified School District. In addition, he was contributing editor of Science World for two years, participated in writing projects for the SMSG, and served on numerous committees concerned with mathematics education.

**Norman E. Steenrod** was born in 1910. He was raised in Dayton, Ohio and attended the University of Michigan, where he received the B. A. degree in 1932. It was as an undergraduate at Michigan, under the influence of the well-known mathematician R. L. Wilder, that he first became involved in topology. He received his Ph.D. from Princeton University in 1936, working under the famous topologist S. Lefschetz.

Professor Steenrod taught at the University of Chicago (1939-1942), at the University of Michigan (1942-1947) and at Princeton University from 1947 until his death in 1971. Almost all his research papers concern algebraic topology.

He has written *The Topology of Fibre Bundles* and co-authored, with S. Eilenberg, *The Foundations of Algebraic Topology*. He has also been editor, from 1948 to 1962, of the *Annals of Mathematics*.