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.
 A: 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.
A: Perhaps the study of metrizable spaces is what you are looking for? A topological space $X$ is metrizable if there exists a metric $d$ on the set $X$ such that the given topology on $X$ is equal to the metric topology generated by $d$.
One way metrizable spaces are studied is by taking a property $P$ of metric spaces, and turning it into a topological property by making use of the existential quantifier. For example, despite the obvious fact pointed out in the comment of @MarkSaving that completeness is not a topological invariant, one can nonetheless define a Polish space to be a topological space $X$ which is homeomorphic to a complete metric space. So, for example, the metric space $(0,1)$ is not complete, but it is Polish, because it is homeomorphic to $\mathbb R$. My other favorite example of a Polish space (maybe everyone's favorite?) is the irrational numbers $\mathbb I = \mathbb R - \mathbb Q$. The study of Polish spaces is an important subtopic in the general study of metrizable spaces.
