All spaces under consideration are supposed to be compact and Hausdorff. For a space X, its hyperspace exp (X) is the set of all non-empty closed subsets of X taken with the Vietoris topology. Although in 1920s, the famous conjuncture: for a non-degenerate Peano continuum X, exp (X) ≈ Q (Q being the Hilbert cube) was launched, decades had passed before the first concrete examples of hyperspaces were discovered. The class of all compact 0-dimensional metric spaces Z is another natural framework for trying to fix the topological types of hyperspaces. Let C_-1 = Ø, C₀ = {1}, C₁ = C (Cantor discontinuum). A sequence of very regular topologically distinct spaces in Z: C₀, C₁, C₂, … , C_n, … is constructed inductively taking C_n to be the space C together with small enough copies of C_n-3 ⊕ C_n-2 interpolated in each of its removed intervals. In 1964, A Pelczynski proved that for each X in Z, having the set of its isolated points everywhere dense, exp(X) ≈ C₂. In 1972, we successed to prove that for each X in Z (excluding the trivial cases of spaces having finite number of isolated points), exp(X) is one of the following spaces: C₁, C₂, C₁ ⊕ C₂, C₃, C₄, C₅, C₇. In the same 1972, R. Schori, D. Curtis and J. West proved the most significant result on hyperspaces, confirming the above mentioned conjuncture.

In 2005, S. Oka (Topology and its Applications, 149, p. 227 - 237) reproves our result from 1972, without referring to our paper. That motivates me for this gripping narrative on hyperspaces.