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

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

2007-12-05

arxiv.org/abs/0712.0786v1

Mathematics

General Topology

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

Scientific paper

Let $CLB_H(X)$ 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 $CLB_H(R^m)$ 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 $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ as long as it does not contain any of the sets $R, [0,\infty), (-\infty,0]$.

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