Hausdorff hyperspaces of $R^m$ and their dense subspaces

Abstract

Let $CL{B}_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CL{B}_H(R^m)$ 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 ${\ell }_2$. Moreover, we investigate the hyperspace $C{L}_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $C{L}_H(R)$ is homeomorphic to the Hilbert space ${\ell }_2(2^{{\aleph }_0})$ as long as it does not contain any of the sets $R,[0,\infty ),(-\infty ,0]$.

21 pages; to appear in J. Math. Soc. Japan