Introduction to Topology 0828533768, 9780828533768

Table of contents :
Signs and Symbols
Table of Contents
Preface
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
GENERAL TOPOLOGY
1. Topological spaces and continuous mappings
2. Topology and continuous mappings of metric spaces. Spaces R^n, S^(n-1) 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 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. Compactifications of topological spaces. Metrization
Further reading
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
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 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 structure of manifold (example)
11. Nondegenerate critical point and its index
12. Describing homotopy type of manifold by means of critical values
Further reading
HOMOLOGY THEORY
1. Preliminary 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