Our purpose in writing this book is to show how topology arose, develop a few of its elements, and present some of its simpler applications.

Topology came to be recognized as a distinct area of mathematics about fifty years ago, and its major growth has taken place within the last thirty years. It is the most vigorous of the newer branches of mathematics and has been producing strong repercussions in most of the older branches. It got its start in response to the needs of analysis (the part of mathematics containing calculus and differential equations). However, topology is not a branch of anidysis. Instead, it is a kind of geometry. It is not an advanced form of geometry such as projective or differential geometry, but rather a primitive, rudimentary form—one which underlies all geometries. A striking fact about topology is that its ideas have penetrated nearly all areas of mathematics. In most of these applications, topology supplies essential tools and concepts for proving certain basic propositions known as existence theorems.

Our presentation of the elements of topology will be centered around two existence theorems of analysis. The first, given in Part I, is fundamental in the calculus and was known long before topology was recognized as a subject. In working out its proof, we shall develop basic ideas of topology.