Elementary Concepts of Topology

Paul Alexandroff

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486607474
• eBook ISBN: 9780486155067
• Topic: Mathematics
• Subtopic: Topology
• Index: Mathematics

Alexandroff's beautiful and elegant introduction to topology was originally published in 1932 as an extension of certain aspects of Hilbert's Anschauliche Geometrie. The text has long been recognized as one of the finest presentations of the fundamental concepts, vital for mathematicians who haven't time for extensive study and for beginning investigators.
The book is not a substitute for a systematic text, but an unusually useful intuitive approach to the basic concepts. Its aim is to present these concepts in a clear, elementary fashion without sacrificing their profundity or exactness and to give some indication of how they are useful in increasingly more areas of mathematics. The author proceeds from the basics of set-theoretic topology, through those topological theorems and questions which are based upon the concept of the algebraic complex, to the concept of Betti groups which binds together central topological theories in a whole and upon which applications of topology largely rest.
Wholly consistent with current investigations, in which a larger and larger part of topology is governed by the concept of homology, the book deals primarily with the concepts of complex, cycle, and homology. It points the way toward a systematic and entirely geometrically oriented theory of the most general structures of space.
First English translation, prepared for Dover by Alan E. Farley. Preface by David Hilbert. Author's Foreword. Index. 25 figures.

III. Simplicial Mappings and Invariance Theorems.

24. If we review what has been said up till now, we see that the discussion has turned around two main concepts: complex on the one hand and topological space on the other. The two concepts correspond to the two interpretations of the concept basic to all of geometry—the concept of geometrical figure. According to the first interpretation, which has been inherent in synthetic geometry since the time of Euclid, a figure is a finite system of (generally) heterogeneous elements (such as points, lines, planes, etc., or simplexes of different dimensions) which are combined with one another according to definite rules—hence, a configuration in the most general sense of the word. According to the second interpretation, a figure is a point set, usually an infinite collection of homogeneous elements. Such a collection must be organized in one way or another to form a geometrical structure—a figure or space. This is accomplished, for example, by introducing a coordinate system, a concept of distance, or the idea of neighborhoods.²⁴

As we mentioned before, in the work of Poincaré both interpretations appear simultaneously. With Poincaré the combinatorial scheme never becomes an end in itself; it is always a tool, an apparatus for the investigation of the “manifold itself,” hence, ultimately a point set. Set-theoretic methods sufficed, however, in Poincaré’s earliest work because his investigations touched only manifolds and slightly more general geometrical structures.²⁵ For this reason, and also in view of the great difficulties connected with the general formulation of the concept of manifold, one can hardly speak of an intermingling or merging of the two methods in Poincaré’s time.

The further development of topology is marked at first by a sharp separation of set-theoretic and combinatorial methods: combinatorial topology had been at the point of believing in no geometrical reality other than the combinatorial scheme itself (and its consequences), while the set-theoretic direction was running into the same danger of complete isolation from the rest of mathematics by an accumulation of more and more specialized questions and complicated examples.

In the face of these extreme positions, the monumental structure of Brouwer’s topology was erected which contained—at least in essence—the basis for the rapid fusion of the two basic topological methods which is presently taking place. In modern topological investigations there are hardly any questions of great importance which are not related to the work of Brouwer and for which a tool cannot be found—often readily applicable —in Brouwer’s collection of topological methods and concepts.

In the twenty years since Brouwer’s work, topology has gone through a period of stormy development, and we have been led—mainly through the great discoveries of the American topologists²⁶—to the present “flowering” of topology, in which analysis situs—still far removed from any danger of being exhausted—lies before us as a great domain developing in close harmony with the most varied ideas and questions of mathematics.

At the center of Brouwer’s work stand the topological invariance theorems. We collect under this name primarily theorems which maintain that if a cer...

Table of contents

• Title Page
• Copyright Page
• TRANSLATOR’S PREFACE
• PREFACE
• FOREWORD
• Table of Contents
• INTRODUCTION
• I. Polyhedra, Manifolds, Topological Spaces.
• II. Algebraic Complexes.
• III. Simplicial Mappings and Invariance Theorems.
• INDEX
• DOVER BOOKS ON MATHEMATICS