Topology - An Invitation

Description

This book starts with a discussion of the classical intermediate value theorem and some of its uncommon “topological” consequences as an appetizer to whet the interest of the reader. It is a concise introduction to topology with a tinge of historical perspective, as the author’s perception is that learning mathematics should be spiced up with a dash of historical development. All the basics of general topology that a student of mathematics would need are discussed, and glimpses of the beginnings of algebraic and combinatorial methods in topology are provided. All the standard material on basic set topology is presented, with the treatment being sometimes new. This is followed by some of the classical, important topological results on Euclidean spaces (the higher-dimensional intermediate value theorem of Poincaré–Miranda, Brouwer’s fixed-point theorem, the no-retract theorem, theorems on invariance of domain and dimension, Borsuk’s antipodal theorem, the Borsuk–Ulam theorem and the Lusternik–Schnirelmann–Borsuk theorem), all proved by combinatorial methods. This material is not usually found in introductory books on topology. The book concludes with an introduction to homotopy, fundamental groups and covering spaces. Throughout, original formulations of concepts and major results are provided, along with English translations. Brief accounts of historical developments and biographical sketches of the dramatis personae are provided. Problem solving being an indispensable process of learning, plenty of exercises are provided to hone the reader's mathematical skills. The book would be suitable for a first course in topology and also as a source for self-study for someone desirous of learning the subject. Familiarity with elementary real analysis and some felicity with the language of set theory and abstract mathematical reasoning would be adequate prerequisites for an intelligent study of the book. Preface References Contents About the Author 1 Apéritif: The Intermediate Value Theorem 1.1 Intermediate Value Theorem 1.2 Biographical Notes 1.2.1 Bolzano References 2 Metric Spaces 2.1 Metrics 2.2 Continuity and Open Sets 2.3 Biographical Notes 2.3.1 Fréchet References 3 Topological Spaces 3.1 Topologies and Open Sets 3.2 Basic Open Sets 3.3 Closed Sets 3.4 Biographical Notes 3.4.1 Riemann 3.4.2 Weyl 3.4.3 Hausdorff 3.4.4 F. Riesz References 4 Continuous Maps 4.1 Limit Points 4.2 Continuity 4.3 Biographical Notes 4.3.1 Cauchy 4.3.2 Weierstrass 4.3.3 Dirichlet References 5 Compact Spaces 5.1 Compactness in mathbbRn 5.2 Compactness in Metric Spaces 5.3 Compactness in Topological Spaces 5.4 Biographical Notes 5.4.1 E. Borel 5.4.2 Heine 5.4.3 Lebesgue 5.4.4 Cantor 5.4.5 Vietoris References 6 Topologies Defined by Maps 6.1 Initial and Final Topologies 6.2 Product Topology 6.3 Quotient Topology 6.4 Biographical Notes 6.4.1 R. L. Moore 6.4.2 Möbius 6.4.3 Klein References 7 Products of Compact Spaces 7.1 Tychonoff's Theorem 7.2 Appendix: Axiom of Choice 7.3 Biographical Notes 7.3.1 Bourbaki 7.3.2 Čech 7.3.3 Tychonoff 7.3.4 Kelley References 8 Separation Axioms 8.1 Hausdorff Spaces 8.2 Normal Spaces 8.3 Regular Spaces 8.4 Completely Regular Spaces 8.5 Biographical Notes 8.5.1 Tietze 8.5.2 Urysohn 8.5.3 Carathéodory 8.5.4 M. H. Stone References 9 Connected Spaces 9.1 Path Connected Spaces 9.2 Connected Spaces 9.3 Locally Connected Spaces 9.4 Biographical Notes 9.4.1 Jordan 9.4.2 Hahn 9.4.3 Kuratowski 9.4.4 Knaster References 10 Countability Axioms 10.1 Countability Properties 10.2 Urysohn Metrisation 10.3 Biographical Notes 10.3.1 Lindelöf References 11 Locally Compact Spaces 11.1 Local Compactness 11.2 One-Point Compactification 11.3 Biographical Notes 11.3.1 Alexandroff 11.3.2 Dieudonné References 12 Complete Metric Spaces 12.1 Completeness and Ascoli–Arzelà 12.2 Bourbaki, Baire and Banach 12.3 Completion 12.4 Biographical Notes 12.4.1 Ascoli 12.4.2 Arzelà 12.4.3 Baire 12.4.4 Banach References 13 Combinatorial Methods in Euclidean Topology 13.1 Convex Sets and Balls 13.2 Cubes and Simplices 13.3 Sperner's Lemma, the Cubical Version 13.4 Poincaré and Brouwer 13.5 Invariance of Domain and Dimension 13.6 Borsuk and the Sphere 13.7 Biographical Notes 13.7.1 Bohl 13.7.2 Hadamard 13.7.3 Ulam 13.7.4 Sperner 13.7.5 Mazurkiewicz 13.7.6 Brouwer References 14 Homotopy 14.1 Retracts and Deformation Retracts 14.2 Homotopy of Maps and Paths 14.3 Biographical Notes 14.3.1 Borsuk References 15 Fundamental Groups and Covering Spaces 15.1 The Fundamental Group 15.2 Examples and Applications 15.3 Covering Spaces 15.4 Biographical Notes 15.4.1 Poincaré References Appendix Appendix: Selected Exercises—Suggestions and Hints Index