Topology

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 PoincaréMiranda, Brouwer's fixed-point theorem, the no-retract theorem, theorems on invariance of domain and dimension, Borsuk's antipodal theorem, the BorsukUlam theorem andthe LusternikSchnirelmannBorsuk 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.

Gives proofs of many deep results by using elementary methods Introduces the Poincare's higher-dimensional intermediate value theorem Presents the recent KulpaTao proofs of domain and dimension invariance

Auteur
K. Parthasarathy is Former Director and Head of the Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India. He earned his doctoral degree from the Indian Institute of Technology Kanpur, after his schooling and college education in Chennai (earlier Madras). His areas of research are abstract harmonic analysis and the theory of frames. He had taught subjects ranging from algebraic number theory to algebraic topology, differential equations to differential geometry and linear algebra to Lie algebras for about 35 years at the postgraduate level at different institutions. He had been Doctoral Adviser for several students and has published a number of research papers in international journals of repute. He is Reviewer for several research journals and for Mathematical Reviews and zbMATH.

Résumé

"This book can and should serve as a template for how introductory texts in other subjects might be 'spiced up' with references to primary historical sources and biographical snippets that allow the reader to explore the material seemingly hand in hand with the mathematical giants who created it." (Don Larson, MAA Reviews, January 4, 2023)

"Clearly set out results and their proofs, illustrated by good examples and followed by a range of exercises ... . The author has also been keeping in touch with recent literature, making use of recent new insights into proofs of old results." (David B. Gauld, zbMATH 1498.54002, 2022)

Contenu
1 Aperitif: The Intermediate Value Theorem.- 2 Metric spaces.- 3 Topological spaces.- 4 Continuous maps.- 5 Compact spaces.- 6Topologies dened by maps.- 7 Products of compact spaces.- 8 Separation axioms.- 9 Connected spaces.- 10 Countability axioms.