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?In this book, my aim is to provide the reader with a foundation in general topology that will adequately prepare him for further work in a broad variety of mathematical disciplines. The arrangement of the material is such that the book can also serve as a reference for the more advanced mathematician.
The reader is assumed to have a background of at least one semester of rigorous analysis; for such persons, the treatment herein is self- contained and the material can easily be covered in a two-semester course.
Chapters I-II provide a short introduction to the axiomatic foundations of set theory. Chapters III-VI are devoted to general topological structures; the emphasis is on the mapping, and an extended treatment of identifications is given. Separation axioms are introduced in Chapter VII; after that, with only very few and explicitly noted exceptions, all spaces in the book are assumed to be Hausdorff. By imposing increasingly more severe conditions on the topology, the development proceeds down the hierarchy of topological spaces to the metric spaces; then convergence, compactness, function spaces, and completeness are taken up. The use of homotopy methods in general topology is started in Chapter XV. In the usual manner, these methods lead to "elementary" proofs of the classical results in Euclidean n-space, such as Brouwer's fixed-point and domain-invariance theorems; a proof of the complete Jordan curve theorem is also given. After discussing the homotopy
classification of spaces and some of its features, the entry of algebra into topology (via the fundamental and higher homotopy groups) is presented in Chapter XIX ; this chapter is part of some work that was done jointly with W. Hurewicz.
The last chapter is devoted to the covering homotopy theorem in fiber spaces and illustrates an interplay between many of the
concepts discussed in the book. Two appendices, one on linear topological spaces and the other on limit spaces, are included.
Nearly every definition is followed by examples illustrating the use of the abstract concept in some fairly concrete situations. This device makes the book suitable for self-study. It also enables the instructor who uses the book as a text to proceed rapidly to those parts of the subject that he deems of greater importance.
Remarks, in small type, call attention either to further developments, or to direct applications in other branches of mathematics.
The problems, which are given at the end of each chapter, can all be solved by the methods developed in the book. Moreover, no proof in the text relies on the solution of any problem. Some of the problems are routine. Others are important theorems that complement the material in the text; these are accompanied by hints for their solution.
The notation of symbolic logic used throughout the book is given immediately after the table of contents.
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