Topology : general & algebraic

176 Pages·2007·1.27 MB·English

by  Chatterjee

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Copyright © 2007, New Age International (P) Ltd., Publishers
Published by New Age International (P) Ltd., Publishers
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Preface v
Preface
Some areas of human knowledge ever since its origin had shaken our understanding of the universe
from time to time. While this is more true about physics, it is true about mathematics as well. The birth
of topology as analysis situs meaning rubbersheet geometry had a similar impact on our traditional
knowledge of analysis. Indeed, topology had enough energy and vigour to give birth to a new culture of
mathematical approach. Algebraic topology added a new dimension to that. Because quantum physicists
and applied mathematicians had noted wonderful interpretations of many physical phenomena through
algebraic topology, they took immense interest in the study of topology in the twentieth century. Indian
physicists too did not lag behind their counterparts in this respect. Some physicists of Kolkata and
around invited me in 1978 to deliver a series of lectures on the subject in the Calcutta University under
the auspices of Satyendra Nath Bose Institute of Physical Sciences. The same lecture was delivered
earlier to the working physicists of the Indian Statistical Institute in 1976. The present manuscript is a
slightly organized version of those lectures delivered at the said places.
To facilitate the readers distinguish the two approaches to the study of topology, matters have been
divided into two parts, viz., general topology and algebraic topology. The general topology introduces
the classical notions of topology such as compactness, completeness, connectedness etc. and the algebraic
topology brings to light the purely algebraic aspects of them. In general, the treatment is sketchy but
motivating and helpful for physicists to grasp quickly the basic ideas. The matters have been tested for
presentation in Shibaji University and Mosul University. The author will feel rewarded if any one studying
this monograph become interested in the subject.
In the preparation of this manuscript I got generous help from many–in particular from Prof A.B.Raha
and Prof H. Sarbadhikari who opted to write a part of the manuscript from lectures. I owe a lot to both
of them. I will be failing in my duty if I do not acknowledge my debt to Prof K. Sikdar, Prof T. Chandra,
Prof S.M. Srivastava, Prof S. Roy, all of Indian Statistical Institute, Prof M. Datta, Director, SNBIPS,
Prof B.K. Datta of the University of Trieste, Prof M.K. Das of the University of Nairobi,
ProfS.Mukhopadhyay of the City University of New York, USA. My last words of gratitude must go
to my wife, Suparna, sons Anandarup and Raju for what they did to see this project complete.
D. CHATTERJEE
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Contents vii
Contents
Preface v
P I
A R T
1. Sets, Relations and Functions 3
1.1 Symbols and Notations 3
1.2 Sets and Set Operations 3
1.3 Relations 11
1.4 Order Relations and Posets 15
1.5 Functions and their Graphs 16
1.6 Countability 23
2. Topologies of R and R2 25
2.1 Topology of R 25
2.2 Continuous Functions and Homeomorphisms 28
2.3 Topology of R2 29
2.4 Continuous Function and Homeomorphism 32
3. Metric Space 33
3.1 Some Definitions 33
3.2 Topology of Metric Spaces 37
3.3 Subspace 41
3.4 Completeness 42
3.5 Continuity and Uniform Continuity 46
3.6 Equivalence, Homeomorphism and Isometry 49
3.7 Compactness 51
3.8 Connectedness 57
viii Contents
4. Topological Spaces 61
4.1 Some Definitions 61
4.2 Neighbourhood, Interior, Exterior and Boundary 62
4.3 Relative Topology and Subspace 64
4.4 Base and Subbase of a Topology 64
4.5 Continuous Functions 66
4.6 Induced Topology 69
4.7 Identification Topology 70
4.8 Free Union of Spaces and Attachments 70
4.9 Topological Invariant 71
4.10 Metrization Problem 71
5. Separation Axioms 72
5.1 The Axioms 72
5.2 Uryshon’s Lemma and Tietze’s Extension Theorem 75
6. Compactness 76
6.1 Some Basic Notions 76
6.2 Other Notions of Compactness 78
6.3 Compactification 78
7. Connectedness 80
7.1 Some Basic Notions 80
7.2 Other Notions of Connectedness 82
P I I
A R T
1. Algebraic Preliminaries 87
1.1 Some Basic Notions 87
1.2 Free Abelian Group 88
1.3 Normal Subgroups 89
1.4 Ideals of Rings 89
1.5 G-Spaces 90
1.6 Category and Functor 92
2. Homotopy Theory 94
2.1 Basic Notions 94
Contents ix
2.2 Homotopy Class 96
2.3 Homotopy Equivalence 96
2.4 Retraction and Deformation 99
2.5 The Fundamental Group 99
2.6 Fundamental Group of the Circle 106
2.7 Lifting Lemma 106
2.8 Covering Homotopy Lemma 106
2.9 The Fundamental Group of a Product Space 108
3. Compact Open Topology 111
3.1 Compact Open Topology on Function Spaces 111
3.2 Loop Spaces 113
3.3 H-Structures 117
3.4 H-Homomorphisms 119
3.5 HOPF Space 120
4. Higher Homotopy Groups 122
4.1 The n-Dimensional Homotopy Group 123
4.2 Homotopy Invariance of the Fundamental Group 126
5. Surfaces, Manifolds and CW Complexes 129
5.1 Surfaces 130
5.2 Manifold 131
5.3 CW Complexes 133
5.4 Fibre Bundles 134
6. Simplicial Homology Theory 135
6.1 Simplex and Simplicial Complex 136
6.2 Triangulation 136
6.3 Barycentric Subdivision 137
6.4 Simplicial Map 138
6.5 Simplicial Approximation 138
6.6 Homology Group 139
6.7 Hurewicz Theorem 141
6.8 Co-Chain, Co-Cycle, Co-Boundary and Co-Homology 141
6.9 Cup Product 142