Topology: An Invitation

This book provides a concise introduction to topology with a historical perspective, covering general topology, Euclidean spaces, and classical results. It includes original formulations, problem solving, and exercises and is suitable for a first course in topology or self-study. Prerequisites include elementary real analysis and familiarity with set theory and abstract mathematical reasoning.

Format: Paperback / softback
Length: 267 pages
Publication date: 10 July 2022
Publisher: Springer Verlag, Singapore

This comprehensive textbook offers a thorough introduction to topology, a branch of mathematics that explores the properties of spaces that remain unchanged under continuous transformations. It begins by discussing the classical intermediate value theorem and its intriguing "topological" consequences, serving as a captivating appetizer to pique the reader's interest. The authors present a concise yet comprehensive overview of topology, incorporating a historical perspective to enhance the learning experience. The book covers all the essential basics of general topology, ensuring that students of mathematics have a solid foundation in the field. Additionally, it provides glimpses into the early development of algebraic and combinatorial methods in topology, offering a unique perspective.

The text proceeds to present a comprehensive treatment of basic set topology, including both familiar and innovative approaches. It covers topics such as the fundamental concepts, properties, and applications of sets, as well as the construction and analysis of topological spaces. The authors introduce various important topological results on Euclidean spaces, which are typically not found in introductory topology books. These results include the higher-dimensional intermediate value theorem of Poincaré-Miranda, Brouwers fixed-point theorem, no-retract theorem, theorems on invariance of domain and dimension, Borsuks antipodal theorem, Borsuk-Ulam theorem, and the Lusternik-Schnirelmann-Borsuk theorem. These results are all proved using combinatorial methods, providing a fresh and exciting perspective on topology.

The book concludes with an introduction to homotopy, fundamental groups, and covering spaces, which are essential topics in topology. Original formulations of concepts and major results are provided, along with English translations, making the material accessible to a wide range of readers. Brief accounts of historical developments and biographical sketches of the key figures in topology add depth and context to the subject. Problem-solving exercises are included throughout the book to reinforce the reader's mathematical skills and encourage independent thinking.

This textbook is suitable for both first-year topology courses and self-study for individuals who are eager to learn the subject. It assumes a basic familiarity with elementary real analysis and some proficiency in linear algebra, making it accessible to students with a strong mathematical background. Whether you are a student, researcher, or enthusiast, this book will provide you with a solid foundation in topology and inspire you to explore its fascinating properties further.

Weight: 493g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811694837
Edition number: 1st ed. 2022