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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.

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    $\begingroup$ Honestly, have you read the wikipedia page on metric spaces before posting? It seems to me that more or less any reasonable property can be found there en.wikipedia.org/wiki/Metric_space and then by browsing through en.wikipedia.org/wiki/Category:Metric_geometry $\endgroup$
    – Theo Buehler
    Commented May 10, 2011 at 16:07
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    $\begingroup$ Metric spaces are an extremely useful abstraction of structure found in many interesting examples. It is better to learn about these examples rather than ask "what can we add to the definition of a metric space to obtain interesting things"? $\endgroup$
    – Yemon Choi
    Commented May 10, 2011 at 19:28
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    $\begingroup$ On looking at that wikipedia entry, I think that the examples given there are worth looking at. Probably the reason why you don't see a mention of Polish spaces is that they are topological spaces that happen to be separable and metrizable; they are not metric spaces per se, because different metrics can give the same topology. Unusurprisingly, Wikipedia's entry on Polish spaces en.wikipedia.org/wiki/Polish_space is a better place to look for those examples. $\endgroup$
    – Yemon Choi
    Commented May 10, 2011 at 19:32
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    $\begingroup$ A POLISH SPACE IS NOT A METRIC SPACE. It is a separable topological space which can be equipped with a complete metric. There are many different metric spaces on $R^2$ which yield the same (usual) topology, and making a list of all of them will tell you very little of worth about the Polish space $R^2$ that is not already obvious from using the "usual" metric $\endgroup$
    – Yemon Choi
    Commented May 10, 2011 at 23:40
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    $\begingroup$ @Marek: Your comment "every complete separable metric space is Polish" suggests (to me) that you missed the point of Yemon's pervious comment. Although what you wrote is true, it's also true that some incomplete separable 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. $\endgroup$
    – Andreas Blass
    Commented May 10, 2011 at 23:45

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Another class of metric spaces that are of interest are length spaces. Roughly speaking, these are spaces in which you can measure the length of paths. The distance is then the inf of all lengths paths pinned at the two points. Gromov talks about these as does a book by Borago and Borago (the exact citations elude me).

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