Given any metric space M, Hausdorff defined a new metric space h(M) whose "points" are the nonempty closed and bounded subsets of M. The hierarchy emerges from the following iteration process. Let H(0,M)=M and for each nonnegative integer n, let H(n+1,M)=h(H(n,M)). If M is (for example) a finite dimensional Euclidean space or Hilbert space, does this process ever reach a fixed pointin the sense that there exists a nonnegative integer k for which the spaces H(k,M) and H(k+1,M) are homeomorphic (or even isometric)? If there is no fixed point in the case of some particular metric space M, is it possible to continue this iteration process into the transfinite?
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