Introduction to Topology

Description

Cover Page PREFACE 1 First Notions of Topology 1. WHAT lS 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 2 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. FACTOR SPACE AND QUOTIENT TOPOLOGY 4. CLASSIFICATION Of SURFACES 5. ORBIT SPACES. PROJECTIVE AND LENS SPACES 6. OPERATIONS OVER SETS IN A TOPOLOOICAL SPACE 7. OPERATIONS OVER SETS IN METRIC SPACES. SPHERE AND BALLS. COMPLETENESS B. 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. COMPACTIFICATIONS OF TOPOLOGICAL SPACES. METRIZATION FURTHER READING 3 Homotopy Theory 1. MAPPING SPACES. HOMOTOPIES. RETRACTIONS, 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 4 ManifoldS and Fibre BUNDLES 1. BASIC NOTIONS OF DIFFERENTIAL CALCULUS IN N-DIMENSIONAL 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. TANGENT VECTOR AS DlFFERENTlAL OPERATOR. DtFFERENTlAL OF FUNCTION AND COTANGENT BUNDLE 8. VECTOR FIELDS ON SMOOTH MANIFOLDS 9. FlBRE BUNDLES AND COVERINGS 10. SMOOTH FUNCTION ON MANIFOLD AND CELLULAR STRUCTURE OF MANIFOLD (EXAMPLE) 11. NONDEGENERATE CRITICAL POINT AND ITS INDEX 12. DESCRIBING HOMOTOPY TYPE OF MANIFOLD BY MEANS OF CRITICAL VALUES FURTHER READING 5 HomologyTheory 1. PRELlMINARY NOTES 2. HOMOLOGY GROUPS OF CHAIN COMPLEXES 3. HOMOLOGY GROUPS OF SIMPLICIAL COMPLEXES 4. SINGULAR HOMOLOGY THEORY 5. HOMOLOGY THEORY AXIOMS 6. HOMOLOGY GROUPS OF SPHERES. DEGREE OF MAPPING 7. HOMOLOGY GROUPS OF CELL COMPLEXES 8. EULER CHARACTERISTIC AND LEFSCHETZ NUMBER FURTHER READING ILLUSTRATIONS REFERENCES NAME INDEX SUBJECT INDEX