Book recommendation for Metric Spaces

Topology of Metric Spaces, Alpha Science International Ltd. (2005) by S. Kumaresan gives a very streamlined development of a course in metric space topology emphasising only the most useful concepts and geometric ideas. To encourage geometric thinking, the book boasts of having a large number of examples to develop our intuition and draw conclusions and generate ideas for proofs. There are a lot of exercises aiding in grasping the material to the core.

Metric Spaces, Springer (2000) by Satish Shirali and Harikrishnan. L. Vasudeva provides a leisurely approach to the theory of metric spaces. To ensure that the ideas take root gradually but firmly, a large number of examples and counterexamples follow each definition. Also included are several worked examples and exercises. Applications of the theory are spread out over the entire book.

Metric Spaces, Springer (2007) by Micheál O'Searcoid is a book from Springer Undergraduate Mathematics Series, provides a fairly thorough introduction to metric spaces. It has a lot of examples and problems related to almost all the basic concepts. Interested ones can also check their answers with the solution manual.

Apt to it's name, Notes on Metric Spaces offered by Juan Pablo Xandari is very helpful with simpler proofs and helpful for quick revisions.

Of course Rudin's Principles of Mathematical Analysis is widely regarded as a standard resource for analysis, though it actually doesn't deal with compactness or connectedness.