Excellent course book
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This is the book that was used in the (now defunct) M435 Topology course at the Open University. Thus I came to know and love this book.
I've taken a look at a few topology texts and they're all more or less difficult to get to grips with. This one is the most accessible of those.
If you need to understand topology for any reason, or you're studying it and it's not on your list of course works, then get it.
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Clear and easy to follow
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This is an excellent introduction to basic analytic topology and metric spaces. The fundamental concepts are clearly presented and the theory is developed so that it is easy to follow; but the book is also concise and compact (not topologically, I think!!).
I doubt that you could find a better starting place if you want to learn about metric spaces or basic analytic topology.
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Clear and rigorous exposition of elementary material.
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Complements nicely the metric and topological spaces course lectured in the first year at Cambridge and develops compactness in the proper manner and finishes cleanly with the Arzela-Ascoli theorem. Perhaps provides far too few applications of the contraction mapping theorem and occasionally hides important and well known results in starred exercises which are generally interesting and well worth doing -- equivalence of norms on R^n, one-point compactification and various counterexamples to unreasonable assumptions of continuity to name just a few.
The author also seems to have completely forgotten the p-adic numbers when developing metric spaces which is a huge loss.
The book largely misses off the more fun aspects of topology which are explored in their beauty in, for example, Henle's "A combinatorial introduction to topology".
Overall, I have yet to find a better book for anyone to begin study of topology from but this book is not enough on its own for the reader to truly enjoy the subject.
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A comprehenisve book that is easy to read
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This is a great book for students taking credits in metric and topological spaces, for example the G13MTS course at Nottingham. It is easy to read, and quite small, yet covers nearly all the introductary material very well. Also contains problems to solve if you want some extra practice.
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Extremely comprehensive self contained text
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Having scouted the shelves at the Nottingham University library, this was the only text I found which gave an introduction into both Metric and Topological Spaces without the prerequisite for deep previous knowledge. The book starts with a recap of some results from real analysis and develops these comprehensively by first looking at properties of Metric spaces and then the more general topological space scenario. It's a very self contained text with good use of examples and counterexamples to assist learning.
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