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on April 11, 2003
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.
I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson�s book was my first serious look at topology.
My reading of Mendelson�s 200-page text required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally of the form �Prove that �.�.
The first chapter provides a concise overview of set theory and functions that is essential for Mendelson�s subsequent set-theoretic analysis of metric spaces and topology.
The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space in a section titled �an infinite dimensional Euclidian space�.
The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.
Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.
He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of �arbitrary closeness� in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.
In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.
Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. I will definitely need to look at another text or two before I can handle more advanced topics.
I suspect that a reader familiar with analysis would have substantially less difficulty with the last two chapters.
In summary, Introduction to Topology quite useful for self-study. Mendelson�s short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.
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This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work.
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on May 15, 2000
I know that some people don't like Dover, but I think Dover is great, and Mendelson's Introduction of Topology is an example of why.
Although the book is very short (around 150 pages), it covers the basics of topology very thoroughly and should prepare the reader for the considerably more abstruse Spanier's Algebraic Topology or other texts of such ilk.
If you are a recreational topologist, or are simply tryinging to figure out which way is up in your first topology course, this is for you.
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on August 3, 2000
since, for some reason, my school didn't offer any topology course, I decided to study topology on my own. It was very fortunate that I found this book in the library. That was right after I took my first analysis course. But I could understand most of the book at that time. After reviewing basic set theory, the author discussed metric spaces, and then he motivates the definition of topological spaces. This is great, I think, becuase many of introductory topology books often give the definition of topological spaces with any motivation. However it is very important to motivate each concept in mathematics especially in introductory level. And this book does this. And as I did, this book is even good for indivisual study. However, you can get almost no geometricl flavor of topology from this book. For example, there is only one section in one chapter in which the author discusses the fumdamental group. Thus, after all this is the best introduction to "point-set topology". So if you don't know almost anything about topolosy, I strongly recommend this book. And one more thing. If you are still wondering if you should buy this one, just look at the price!
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