Page 4 - Elementary Concepts of Topology

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 with elementary intuition even in
the most abstract questions, but in doing so I have never given up the
full rigor of the definitions. On the other hand, in the many examples I have
nearly always dispensed with the proofs and been content with a mere
indication of the state of affairs which the example under consideration
served to illustrate.
Mindful of this latter end, I have picked out of the extensive subject
matter of modern topology only one set of questions, namely those which
are concentrated on the concepts of complex, cycle and homology; in
doing so I have not shied away from treating these and related questions
in the full perspective appropriate to the modern state of topology.
With respect to the basis for the choice of materials appearing here, I
have included a paragraph (46) at the end of this book.
Of course, one cannot learn topology from these few pages; if however,
one gets from them some idea of the nature of topology—at least in one
of its most important and applicable parts, and also acquires the desire for
further individual study—then my goal will have been reached. From this
point of view let me direct those of you who already have the desire to
study topology to the book written by Herr Hopf and myself which will
soon be printed by the same publisher [see footnote 4—A.E.F.].
I should like to express my warmest thanks to S. Cohn-Vossen and
O. Neugebauer, who have read this book in manuscript form as well as in
proof and have given me worthwhile advice on many occasions.
My sincere thanks also to Mr. Ephrämowitsch at Moscow and Mr.
Singer at Princeton, who most kindly undertook the drawing of the figures.
P. ALEXANDROFF
Kljasma at Moscow,
May 17, 1932.