Essentials of Topology with Applications (Textbooks in Mathematics) 1st Edition by Steven G. Krantz (PDF)

Description

Brings Readers Up to Speed in This Important and Rapidly Growing AreaSupported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. It presents the traditional concepts of topological space, open and closed sets, separation axioms, and more, along with applications of the ideas in Morse, manifold, homotopy, and homology theories.After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. He also explores meaningful applications in a number of areas, including the traveling salesman problem, digital imaging, mathematical economics, and dynamical systems. The appendices offer background material on logic, set theory, the properties of real numbers, the axiom of choice, and basic algebraic structures.Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I was looking forward to reading this book and had high hopes that it would be a suitable text for our first year graduate course sequence in topology, since Professor Krantz has a reputation for being an excellent expositor of mathematics.After going through about half the book however, I must say that I am seriously disappointed in this book. It seems to be poorly conceived and poorly executed. There are just too many topics covered in this book and most of them are covered very superficially.Let me just give a small random sample of flaws I found.1) The section of Morse theory in Chapter 2 is very cursory, hardly enough to give any useful insight into this topic. Moreover it refers to notions which are not defined (e.g. 1-cell, 2-cell) or to notions which are defined later (e.g. homotopy, defined in Chapter 3)2) In Chapter 3 the author purports to cover singular homology theory without any mention of excision (or equivalently Mayer-Vietoris)!3) In Chapter 4 the torus and Klein bottle are described as being obtained by pasting the sides of a rectangle. This however is never related to the notion of quotient space, which is treated very briefly in Chapter 2, and which notion is needed to rigorously describe what “pasting” really means.This is by no means an exhaustive list, but should be enough to indicate the issues which plague this book.I am not sure what audience this is aimed at. It seems to be a thoughtless jumble of formal arguments interspersed with a lot of informal “hand-waving”.So I am sorry to say that I can not recommend this book as a text in any of our courses.

⭐The star rating and following comments are based only on a cursory reading of the appendices. The topics are presented in fairly clear language and sequence, but there were distracting editing errors. For example, the appendix on Principles of Set Theory repeats a definition for algebraic numbers on two different pages with two different definition numbers. In the same appendix, definition numbers that can’t be found are referenced.

⭐Thankful to you

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