## FOREWORDThis 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.]. I should like to express my warmest thanks to S. Cohn-Vossen and 0. 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. Ephramowitsch at Moscow and Mr. Singer at Princeton, who most kindly undertook the drawing of the figures. P. Alexandroff
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