Hausdorff hyperspaces of $\mathbf{R}^{m}$ and their dense subspaces

KUBIŚ, Wiesław ; SAKAI, Katsuro

J. Math. Soc. Japan, Tome 60 (2008) no. 1, p. 193-217 / Harvested from Project Euclid

Résumé

Let $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ be the hyperspace of nonempty bounded closed subsets of Euclidean space $\mathbf{R}^{m}$ endowed with the Hausdorff metric. It is well known that $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ 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 $0 \leq 1 < m$ , let ¶ $\nu^{m}_{k} = \{x = (x_{i})_{i=1}^{m} \in \mathbf{R}^{m} : x_{i} \in \mathbf{R}\setminus\mathbf{Q}$ except for at most $k$ many $i \}$ , ¶ where $\nu^{2k+1}_{k}$ is the $k$ -dimensional Nöbeling space and $\nu^{m}_{0} = (\mathbf{R}\setminus\mathbf{Q})^{m}$ . It is also proved that the spaces $\mathrm{Bd}_{H}(\nu^{1}_{0})$ and $\mathrm{Bd}_{H}(\nu^{m}_{k})$ , $0\leq k

Détail

Publié le : 2008-01-15
Classification:  the hyperspace,  the Hausdorff metric,  bounded closed sets,  nowhere dense closed sets,  perfect sets,  Cantor sets,  Lebesgue measure zero,  Euclidean space,  Nöbeling space,  the Hilbert cube,  the pseudo-interior,  Hilbert space,  54B20,  57N20