# Elementary Concepts in Topology

## Paul Alexandroff

Language: English

Pages: 62

ISBN: 2:00284806

Format: PDF / Kindle (mobi) / ePub

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.

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rarely has an initially unpromising branch of a theory turned out to be of such fundamental importance for such a great range of completely different fields as topology. Indeed, today in nearly all branches of analysis and in its far-reaching applications, topological methods are used and topological questions asked. Such a wide range of applications naturally requires that the conceptual structure be of such precision that the common core of the superficially different questions may be

rarely has an initially unpromising branch of a theory turned out to be of such fundamental importance for such a great range of completely different fields as topology. Indeed, today in nearly all branches of analysis and in its far-reaching applications, topological methods are used and topological questions asked. Such a wide range of applications naturally requires that the conceptual structure be of such precision that the common core of the superficially different questions may be

recognized. It is not surprising that such an analysis of fundamental geometrical concepts must rob them to a large extent of their immediate intuitivenessâ€”so much the more, when in the application to other fields, as in the geometry of our surrounding space, an extension to arbitrary dimensions becomes necessary. While I have attempted in my Anschauliche Geometric to consider spatial perception, here it will be shown how many of these concepts may be extended and sharpened and thus, how the

recognized. It is not surprising that such an analysis of fundamental geometrical concepts must rob them to a large extent of their immediate intuitivenessâ€”so much the more, when in the application to other fields, as in the geometry of our surrounding space, an extension to arbitrary dimensions becomes necessary. While I have attempted in my Anschauliche Geometric to consider spatial perception, here it will be shown how many of these concepts may be extended and sharpened and thus, how the

FOREWORD THIS LITTLE book is intended for those who desire to obtain an exact idea of at least some of the most important of the fundamental concepts of topology but who are not in a position to undertake a systematic study of this many-sided and sometimes not easily approached science. It was first planned as an appendix to Hilbert's lectures on intuitive geometry, but it has subsequently been extended somewhat and has finally come into the present form. I have taken pains not to lose touch