We now come to two volume set on Introduction to Topology by Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko.

This is a two volume book set, which has 5 sections in all. It is based on lectures delivered by Yu. G. Borisovich at Mathematics Department of Voronezh University. Each of
the section is preceeded by an illustration which has a lot of mathematical content which are by Prof. A. T. Fomenko.

About the book

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.

Thanks to the original uploader for the scan.

Note: The scan quality is poor and text is barely readable at
times. We have added OCR, which is not reliable, reduced file size, combined 2 pdfs into
one, bookmarked and paginated and pdfs.  Vol. 1 and Vol. 2 which are
physically separate books have been combined in one single pdf. The
page numbering is continuous between the two volumes. Page 147 onwards are
the contents of Vol. 2. We have access to the hard copies, and might see a better version in the future.

PDF | Bookmarked | Paginated | OCR | Cover | 324 Pages |  24.2 MB

The Internet Archive Link

Update 02 Jan 2020

 Contents

FIRST NOTIONS OF TOPOLOGY

1. What is topology? 11
2. Generalization of the concepts of space and function 15
3. From a metric to topological space 18
4. The notion of Riemann surface 28
5. Something about knots 34
Further Reading 37

GENERAL TOPOLOGY

1. T opological spaces and continuous mappings 41
2. Topology and continuous mapping of metric spaces. Spaces R^n,
S^(n-1) and D^n 46
3. Factor space and quotient topology 52
4. Classification of Surfaces 57
5 . Orbit Spaces. Projective and Iens spaces 67
6. Operations over sets in topological space 70
7. Operations over sets in metric spaces. Spheres and
balls. Completeness 73
8. Properties of continuous Mappings 76
9. Products of topological spaces 80
10. Connectedness of topological spaces 84
11. Countability and separation axioms 88
12. Normal spaces and functional separability 92
13. Compact spaces and their mappings 97
14. Compactifications of topological spaces. Metrization 105
Further Reading 107

HOMOTOPY THEORY

1. Mapping space. Homotopies, retractions, and deformations 111
2. Category, functor and algebraization of topological problems 118
3. Functors of homotopy grous 121
4. Computing the fundamental homotopy groups of some space 131
Further Reading 146

MANIFOLDS AND FIBRE BUNDLES

1. Basic notions of differential calculus in n-dimensional space 149
2. Smooth submanifolds in Euclidean space 157
3. Smooth Manifolds 161
4. Smooth functions in a manifold and smooth partition of unity 173
5. Mapping of manifolds 180
6. Tangent bundle and tangential map 188
7.  Tangent vector as differential operator. Differential of function
and cotangent bundle 199
8. Vector fields on smooth manifolds 208
9. Fibre  bundles and covering  213
10. Smooth function on manifold and cellular structure of manifold (example) 235
11. Non-degenrate ciritcal point and its index 240
12. Describing homotopy type of manifold by means of critical values 244
Further Reading 249

HOMOLOGY THEORY

1. Preliminary Notes 253
2. Homology groups of chain complexes 255
3. Homology groups of simplical complexes 257
4. Singular Homology Theory 268
5. Homology theory axioms 278
6. Homology groups of spheres. Degree of mapping 281
7. Homology groups of cell complexes 289
8. Euler Characteristic and Lefschetz number 292
Further reading 299

Illustrations 302

References 303

Name Index 306

Subject Index 308