Description
Book SynopsisAn Illustrated Introduction to Topology and Homotopy explores the beauty of topology and homotopy theory in a direct and engaging manner while illustrating the power of the theory through many, often surprising, applications. This self-contained book takes a visual and rigorous approach that incorporates both extensive illustrations and full proofs.
The first part of the text covers basic topology, ranging from metric spaces and the axioms of topology through subspaces, product spaces, connectedness, compactness, and separation axioms to Urysohn's lemma, Tietze's theorems, and Stone-Cech compactification. Focusing on homotopy, the second part starts with the notions of ambient isotopy, homotopy, and the fundamental group. The book then covers basic combinatorial group theory, the Seifert-van Kampen theorem, knots, and low-dimensional manifolds. The last three chapters discuss the theory of covering spaces, the Borsuk-Ulam theorem, and applications in gr
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"… reflects interesting aspects and will find its readers."—Zentralblatt MATH, 1323
"… an ideal college or university textbook and an invaluable addition to academic library mathematical studies reference collections."
—Library Bookwatch, May 2015
Table of ContentsTOPOLOGY: Sets, Numbers, Cardinals, and Ordinals. Metric Spaces: Definition, Examples, and Basics. Topological Spaces: Definition and Examples. Subspaces, Quotient Spaces, Manifolds, and CW-Complexes. Products of Spaces. Connected Spaces and Path Connected Spaces. Compactness and Related Matters. Separation Properties. Urysohn, Tietze, and Stone-Čech. HOMOTOPY: Isotopy and Homotopy. The Fundamental Group of a Circle and Applications. Combinatorial Group Theory. Seifert-van Kampen Theorem and Applications. On Classifying Manifolds and Related Topics. Covering Spaces, Part 1. Covering Spaces, Part 2. Applications. Applications in Group Theory. Bibliography.
