**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?).



Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube
The hyperspace of any Peano continuum (locally connected metric continuum) is homeomorphic to the Hilbert cube; this is a result of Curtis and Schori, [see here](http://www.ams.org/journals/bull/1974-80-05/S0002-9904-1974-13579-2/S0002-9904-1974-13579-2.pdf).
I learnt about this result in a paper of Torunczyk, where a different proof is given ([that paper is also available online)](http://matwbn.icm.edu.pl/ksiazki/fm/fm106/fm10614.pdf)


When it comes to asking about an isometry, I'm not sure which metric you put on the Hilbert cube...


