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.
Author(s): K. Parthasarathy
Series: UNITEXT, 134
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 267
City: Singapore
Tags: Topology, Metric Space, Compactness, Separation Axioms, Connectedness, Homotopy, Fundamental Group, Covering Space
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
