A good book for metric spaces?

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?

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.

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.

Good introduction to Metric Spaces Metric Spaces: Iteration and Application

Rudin, Principles of Mathematical Analysis.

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.