What are some advanced books on metric space? Metric space, with the additional notion of “distance between points”, has properties that are more “concrete” than a topological structure. After a basic study I saw a number of strange and interesting results which depend heavily on the metric structure. So I wonder if there is any advanced book on metric space after learning general topology, supposing that one is familiar with the metrizability theorems and have a basic knowledge on metric structure, say, up to Willard’s General Topology. Any recommendation would be helpful. Thank you.

I can recommend the following books which deal mainly with metric spaces:


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*Heinonen: Lectures on Analysis in Metric spaces This is a short and well written book talking about various topics and well suited for self-study.

*Bridson, Martin R., Häfliger, André : Metric Spaces of Non-Positive Curvature A more advanced book dealing with non-positively curved spaces. CAT($\kappa$) spaces and (Gromov) $\delta$-hyperbolic spaces. It is kind of the standard reference for those spaces.

*Stephanie Alexander, Vitali Kapovitch, Anton Petrunin: Invitation to Alexandrov geometry Dealing with CAT(0) spaces. Freely available online.

*S. Alexander, V. Kapovitch, A. Petrunin Alexandrov Geometry Spaces with curvature bounded from below.

