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I'm looking for a book to study metric spaces. Two years ago, I used a book written by Burkill. While using multiple topological concepts, I studied Munkres (chapters 2, 3, 4, 5, 6 & 9). I do not feel as comfortable in group theory, ring theory or general topology, so I feel that books such as those written by Hungerford, Robinson, and Atiyah and Mcdonald have been very important in my education. But I have not had that feeling with any book of analysis / metric spaces. I feel that my knowledge in this area is still very basic. What book would you recommend to study analysis?

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There is a wonderful short book by Kaplansky called Set Theory and Metric Spaces.

How good is it? I was able to read it at the beginning of my undergraduate career, and in the intervening years of undergraduate study, graduate study and then mathematical research, I have only ever turned to other books on either of these subjects out of idle curiosity: everything I have ever needed to know is in Kaplansky's text.

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    $\begingroup$ I'll second the recommendation on Kaplansky's text. It's very easy to read and is inexpensive; a perfect first book on Metric Spaces. $\endgroup$ – ItsNotObvious Dec 8 '11 at 17:00
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    $\begingroup$ Kaplansky was one of the great master teachers at the University of Chicago in it's heyday and it's very sad most of his lecture notes are out of print and very expensive. $\endgroup$ – Mathemagician1234 Jan 21 '12 at 0:10
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A good book for metric spaces specifically would be Ó Searcóid's Metric Spaces. However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis.

A good book for real analysis would be Kolmogorov and Fomin's Introductory Real Analysis.

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Rudin, Principles of Mathematical Analysis.

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Good introduction to Metric Spaces Metric Spaces: Iteration and Application

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Dieudonné's Foundations of modern analysis Chapter 3 is a very thorough treatment of metric spaces. It's a bit dry, but perfect as reference.

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$Spaces$ by Tom Lindstrøm. It’s a new book and is published by the AMS, but it’s $\textbf{very}$ good. Chapter 3 covers metric spaces. Chapters 4, 5, 6, 7, 8, and 9 cover spaces of continuous functions, normed spaces and linear operators, calculus in normed spaces, measure and integration, constructing measure, and Fourier series, respectively. (Of course baby Rudin is also a neccessity :p.)

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A good and concise book would be "Metric and Topologial Spaces" by Sutherland.

Note that metric and topological spaces are highly related.

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