Hausdorff hyperspaces of R m and their dense subspaces

Abstract

Let ${\mathrm{Bd}}_H(R^m)$ be the hyperspace of nonempty bounded closed subsets of Euclidean space $R^m$ endowed with the Hausdorff metric. It is well known that ${\mathrm{Bd}}_H(R^m)$ is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of ${\mathrm{Bd}}_H(R^m)$ 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 ${\ell }_2$. For each , let

${\nu }_k^m=\{x=(x_i{)}_{i=1}^m\in R^m:x_i\in R\setminus Q$ except for at most $k$ many $i\}$,

where ${\nu }_k^{2k+1}$ is the $k$-dimensional Nöbeling space and ${\nu }_0^m=(R\setminus Q{)}^m$. It is also proved that the spaces ${\mathrm{Bd}}_H({\nu }_0^1)$ and ${\mathrm{Bd}}_H({\nu }_k^m)$, , are homeomorphic to ${\ell }_2$. Moreover, we investigate the hyperspace ${\mathrm{Cld}}_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component $\mathit{H}$ of ${\mathrm{Cld}}_H(R)$ is homeomorphic to the Hilbert space ${\ell }_2(2^{{\aleph }_0})$ of weight $2^{{\aleph }_0}$ in case where $\mathit{H}$ ∌ $R,[0,\infty ),(-\infty ,0]$.