TITLE: Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube
QUESTION [5 upvotes]: Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$?
Remarks and definitions: 
1) The Hilbert cube $H$ is a compact metric space, where the metric is given by the $\ell_2$-norm of sequences. A classical theorem on metric spaces says that every compact metric space is isometric to a closed subspace of $H$. 
2) The hyperspace of a metric space $X$ is the metric space of all non-empty compact subsets of $X$ given by the Hausdorff metric. Another classical theorem on metric spaces says that the hyperspace of a compact metric space is again a compact metric space.
Combining 1) and 2) shows that the hyperspace of the Hilbert cube is isometric to a closed subspace of the Hilbert cube. So my question asks whether we also can get a homeomorphism 
(can we even get both spaces isometric?).

REPLY [3 votes]: Rule out isometry ... In the hyperspace of $\mathbb R$, let $A=\{0,1,2\}$, $B=\{0\}$, $C=\{2\}$ and $M = \{1\}$.  Then points $A, B, C$ all have distance $2$ from each other; they are the vertices of an equilateral triangle.  But $M$ has distance $1$ from each of $A, B, C$, so $M$ is the midpoint of each of the three sides of that triangle.  Not possible in Euclidean geometry.