Let be the hyperspace of nonempty bounded closed subsets of Euclidean space endowed with the Hausdorff metric. It is well known that is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space . For each , let
except for at most many ,
where is the -dimensional Nöbeling space and . It is also proved that the spaces and , , are homeomorphic to . Moreover, we investigate the hyperspace of all nonempty closed subsets of the real line with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component of is homeomorphic to the Hilbert space of weight in case where ∌ .
Wiesław KUBIŚ. Katsuro SAKAI. "Hausdorff hyperspaces of and their dense subspaces." J. Math. Soc. Japan 60 (1) 193 - 217, January, 2008. https://doi.org/10.2969/jmsj/06010193