Customer Reviews

Introduction to Topology

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Written for undergraduate students of mathematics, this book serves as a fine introduction to topology from an abstract set-theoretic point of view. The approach of the author is also to have the reader do most of the proofs of the theorems in the book, and thus the book can be thought of as the student's second course in proof theory, the first being maybe Euclidean geometry.

Basic concepts in set theory are outlined in chapter 1, and in detail. This paves the way for a discussion of topological spaces in chapter 2. The author could have begun this discussion with a general definition of a topological space, but instead chooses (thankfully) to motivate the definition via the definition of an open set in the real line. The abstract definition is thus better appreciated, which the author then does immediately. He then moves on to the consideration of subspaces and continuity in chapter 3. The discussion here is pretty standard, as there are not any examples that cannot be found in the literature. The most important concept introduced here is that of a homeomorphism, and readers will get a taste of the intuitive "coffee cup = donut" definition of topological equivalance that they have no doubt heard about from popular discussions of topology.

Product spaces are introduced in chapter 4, with a brief peek at topological groups given. Infinite products are introduced but the reading is labeled as supplementary by the author. Chapter 5 is then an introduction to the topological concept of connectedness. The beginner may be troubled as to the way connectedness is defined, since it is defined as the negation of disconnectedness, but the examples given should alleviate any skepticism as to this nonconstructive definition of connectedness. The famous example of a connected space that becomes totally disconnected after the removal of one point is unfortunately not discussed in this book.

Another important concept in topology, that of compactness, is discussed in chapter 6. It is introduced via the concept of coverings, and it is shown that the use of this concept, and not one that is based on a generalization of closed and bounded sets, is the one that gives the best definition for general topological spaces. Then, in chapter 7, the separation properties of topological spaces are discussed. Regular and normal spaces are defined here also, along with the concept of a T5 space. The latter is usually not discussed in elementary books in topology. Metric spaces are finally introduced in the last chapter of the book, giving the reader some of the tools needed for a future study of analysis.

This book is so attainable to a reader with a solid foundation in mathematics regardless of the complexity and depth of the subject matter. Baker first gives a good review of basic set theory, which I believe is always something that can never be reviewed enough. He then proceeds to beautifully bring together definitions and examples leading, of course, to theorems and corrollaries. This is all backed by exercises that, which do require much work and thought, are very reinforcing and rewarding. I have just begun my graduate career in mathematics and having Baker's book at hand has been a saving grace. Although this book is introductory and therefore lacking in some concepts, it covers the basic material so well that moving forward in the subject of topology becomes accesible and enjoyable.

There are no words to describe how much I hate this book. When you have to scour the internet for other books and lectures on EVERY SINGLE topic in a book then that is the definition of an awful book. As a previous reviewer said none of the exercise proofs are worked out and when they are in the lesson they are done in a very confusing manner. One theorems proof had five iff's in it. You literally have to have a Masters or a PHD in math to understand this "great book for undergraduate math students"

I read the chapters and look at the problems and then I immediately begin to scan Google books for other books on the same topic. The problem is that a lot of the notation in this book is exclusive to Mr. Baker so for those exercises you are left hung out to dry.

If you like this book then you already know how to do everything in it. If you want to learn topology and/or proofs then buy another book. If your class requires this book, petition to have it changed.

This books is terrible. There is no answers in the book! You have no way of knowing if you are doing the problems right or not. Also, the examples are too brief and don't explain what steps or theorems that they used. I am buying another book just to figure out what this book is talking about.