Now, for it depends how precisely you define 'interesting'. If you mean just 'cool' you could recall (a proof can be found on my blog here) that the set of all finite subsets of an infinite set is equipotent to the set itself. Thus, is countably infinite. In particular, we can find a bijection . We can then take the -adic metric on and define a metric on to be exactly such that is an isometry. You could even do the same thing but giving the usual metric. Does that work, or do you want something more natural?