Is there any book dealing with properties of a metric space which depend on both the topology and metric structure? Normal books like Munkres tells us how the purely topological properties of metric spaces are preserved under homeomorphisms. But, I wish to know, is there any book which studies invariant properties of metric space which depend on both the topological and metric structure at the same time?
So, for instance, I would wish to know if a property like completeness is preserved under topologically equivalent metric spaces.
Edit: I am not sure why I am getting downvoted or why people think this is covered in a general Analysis text. I have checked Tao analysis-1 and 2, checked Kreyszig and currently reading Munkre's but I have not found in any of these book the topic I wish to check (at least a direct presentation).
If I am to be even more elaborate, it would be like so: I am looking for theorems on topological properties which translate into special metric properties but doesn't go backwards.
So, for example a properties like boundedness or Cauchy sequences.

One can think about the whole analysis as studying of the Euclidean metric and its variations. Mostly. This also opens doors to geometry and differential analysis. So any book on analysis will be good.
Another important object to study in this context is isometries: these depend on the metric, not on topology only.
Another context where metrical properties are studied a lot is data analysis.

So, for instance, I would wish to know if a property like completeness is preserved under topologically equivalent metric spaces.

It is not. $\mathbb{R}$ and $(0,1)$, with the Euclidean metric, are homeomorphic. But $(0,1)$ is not complete with respect to the Euclidean metric.
