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Consider a complete metric space $(M, d)$ and let $F(M)$ denote the non-empty compact subsets of $M$. Then $F(M)$ is also a complete metric space under the Hausdorff distance $d_H$. Given some complete metric space $M$ let us denote this metric space of compact subsets by $M'$. I was wondering about the behavior of the sequence $$M, M', (M')', ...$$

For which metric spaces does sequence terminate in a space $N$ with $N'$ isometric to $N$? For which metric spaces does the sequence not terminate? Is there ever any "monotonicity" in the sequence with respect to isometric embeddings, i.e., when is it true that $M$ is isometrically embedded in $M'$ and so forth? Answers to any of these (or related) questions or references would be greatly appreciated.

Unfortunately, I know next to nothing about these concepts and the question just randomly popped into my head while browsing wikipedia so I have not really had any of my own thoughts about the question. If it is easy, then just a hint would be appreciated. Also, anyone who can think of a more descriptive but succinct title should feel free to change it :)

EDIT: I have noticed that if $M$ is finite (and therefore has the discrete metric) and contains two or more elements, then every subset is compact and $M' = 2^M$ is of strictly larger cardinality than $M$, but still finite. Thus the sequence cannot terminate.

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Do you define the empty set to be $1$ away from everything else? $\:$ – Ricky Demer May 2 '12 at 0:40
@RickyDemer Sorry, the empty set is meant to be excluded. I will edit accordingly. – user12014 May 2 '12 at 1:15
Isn't $M$ isometrically embeddable in $M'$ always as the set of one-point subsets of $M$? – Mariano Suárez-Alvarez May 2 '12 at 1:18
@MarianoSuárez-Alvarez Ah, yes clearly. So at least that part was a stupid question after all. – user12014 May 2 '12 at 1:19
Here is a paper that could be of interest. – user30416 May 2 '12 at 2:56

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