As is often the case in mathematics there is an option of studying a more general topic but this comes with a price of losing some interesting properties which are only present in the more specialized area.

In this case, I am interested in learning about relative usefulness of a metric space carrying some additional structure. To give some obvious examples:

**Polish space**(a separable and completely metrizable space) is very useful in measure and probability theory**Banach space**(a complete normed vector space) is a cornerstone of a functional analysis

I want to learn about more examples of this kind (not necessarily as important though) to get a better idea about how different kinds of metric spaces look like and also about the fields that are based on them.

1. What are some other structures one might impose on a metric space to obtain interesting classes of spaces?

In a related spirit

2. Are there properties (e.g. completeness, separability) that are so important that they are almost always required in some application (or even some field)? If so, what is that application, what are those properties and why are they important (or equivalently, what fails when they are not present)?

*Note:* sorry, if this is too vague or too basic. But I don't have a solid mathematical background, so it's hard to know where to look for answers to these questions. I am just trying to learn about topology in general and about metric spaces in particular (mainly because of their applications in probability theory) and I will definitely accept as an answer a reference to a standard literature.

incompleteseparable metric spaces are Polish (because the same topology can be induced by a different, complete metric). The property of being Polish is a matter of topology; even if you're given a topology as induced by a metric, you may need to change the metric (but not the topology) to see that your space is Polish. – Andreas Blass May 10 '11 at 23:45