This textbooks is one of the many possible variants of a first course in topology and is written in accordance with both the author’s preferences and their experience as lecturers and researchers. It deals with those areas of topology that are most closely related to fundamental courses in general mathematics and applications. The material leaves a lecturer a free choice as to how he or she may want to design his or her own topology course and seminar classes.
The books were translated from the Russian by Oleg Efimov and was
first published by Mir Publishers in 1980.
Author(s): Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko
Publisher: Mir Publisher
Year: 1985
Language: English
Pages: C, 316
Cover
S Title
Introduction to Topolöogy
Copyright (C) 1985 by Mir Publisher
CONTENTS
PREFACE
Chapter I First Notions of Topology
1. WHAT IS TOPOLOGY?
2. GENERALIZATION OF THE CONCEPTS OF SPACE AND FUNCTION
3. FROM A METRIC TO TOPOLOGICAL SPACE
4. THE NOTION OF RIEMANN SURFACE
5. SOMETHING ABOUT KNOTS
FURTHER READING
Chapter II General Topology
1. TOPOLOGICAL SPACES AND CONTINUOUS MAPPINGS
2. TOPOLOGY AND CONTINUOUS MAPPINGS OF METRIC SPACES. SPACES R^n , S^(n-1) , AND D^n
3. FACfOR SPACE AND QUOTIENT TOPOLOGY
4. CLASSIFICATION OF SURFACES
5. ORBIT SPACES. PROJECTIVE AND LENS SPACES
6. OPERATIONS OVER SETS IN A TOPOLOGICAL SPACE
7. OPERATIONS OVER SETS IN METRIC SPACES. SPHERES AND BALLS. COMPLETENESS
8. PROPERTIES OF CONTINUOUS MAPPINGS
9. PRODUCTS OF TOPOLOGICAL SPACES
10. CONNECTEDNESS OF TOPOLOGICAL SPACES
11. COUNTABILITY AND SEPARATION AXIOMS
12. NORMAL SPACES AND FUNCTIONAL SEPARABILITY
13. COMPACT SPACES AND THEIR MAPPINGS
14. COMPACTIFlCATJONS OF TOPOLOGICAL SPACE!S, METRIZATION
FURTHER READING
Chapter III Homotopy Theory
1. MAPPlNG SPACES. HOMOTOPIES ,RETRACTlONS, AND DEFORMATIONS
2. CATEGORY, FUNCTOR AND ALGEBRAIZATION OF TOPOLOGICAL PROBLEMS
3. FUNCTORS Of HOMOTOPY GROUPS
4. COMPUTING THE FUNDAMENTAL AND HOMOTOPY GROUPS OF SOME SPACES
FURTHER READING
Chapter IV Manifolds and Fibre Bundles
1. BASIC NOTlONS OF DIFFEREl'ITIAL CALCULUS lN n-DJMENSlONAL SPACE
2. SMOOTH SUBMANIFOLDS IN EUCLIDEAN SPACE
3. SMOOTH MANIFOLDS
4. SMOOTH FUNCTIONS IN A MANIFOLD AND SMOOTH PARTITION OF UNITY
5. MAPPINGS OF MANIFOLDS
6. TANGENT BUNDLE AND TANGENTIAL MAP
7. TANOEl'IT VECTOR AS DIFFERENTIAL OPERATOR. DIFFERENTIAL OF FUNCTION AND COTANGENT BUNDLE
8. VECTOR FIELDS ON SMOOTH MANIFOLDS
9. FIBRE BUNDLES AND COVERINGS
10. SMOOTH FUNCTION ON MANIFOLD AND CELLULAR STRUCURE OF MANIFOLD (EXAMPLE)
11. NONDEGENERATE CRJTICAL POINT AND ITS INDEX
12. DESCRIBING HOMOTOPY TYPE OF MANIFOLD BY MEANS OF CRITICAL VALVES
FURTHER READING
Chapter V Homology Theory
1. PRELJMJNARY NOTES
2. HOMOLOGY GROUPS OF CHAlN COMPLEXES
3. HOMOLOGY GROUPS OF SIMPLICIAL COMPLEXES
4. SINGULAR HOMOLOGY THEORY
5. HOMOLOOY THEORY AXIOMS
6. HOMOLOGY GROUPS OF SPHERES. DEGREE OF MAPPING
7. HOMOLOGY GROUPS OF CELL COMPLEXES
8. EULER CHARACI'ERISTIC AND LEFSCHETZ NUMBER
FURTHER READING
ILLUSTRATIONS
REFERENCES
NAME INDEX
SUBJECT INDEX
