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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.

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  • $\begingroup$ A particular nice metric is a Riemannian metric. For books, see here. $\endgroup$ – Dietrich Burde Dec 9 '18 at 19:30
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There's a lot of work done on metric spaces in an extended sense, so this might be a bit of too broad a question, but here are some areas that I have come across (but alas, only briefly looked through):

Metric geometry. Found in A Course in Metric Geometry by Burago, Burago, Ivanov and Metric Structures by Gromov. This looks, amongst other things, at metric spaces from infinity, and convergence of metric spaces in the Gromov-Hausdorff convergence, and also their applications to Riemann geometry.

Embeddings of Metric Spaces. This area is focused on trying to embed metric spaces into other spaces, like $\mathbb{R}$, with minimum distortion. There is also other results here like dimensionality reduction on metric spaces. A reference is Matousek's Lecture Notes on Metric Embeddings.

Calculus on Metric Spaces. I'm less sure about this but I think on metric measure spaces, you can do a lot of calculus. A reference I seen is Ambrosio, Gigli, Savare Gradient Flows in Metric Space and the Space of Probability Measures, but I don't claim to understand anything here. Another thing to look for here is Cheeger's result that says that you do differentiable calculus on metric spaces that are not too bad.

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

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