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

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 $R, [0,\infty), (-\infty,0]$.

💡 Research Summary

The paper investigates the topology of Hausdorff hyperspaces of closed bounded subsets of metric spaces, focusing on two main settings: the hyperspace of ℝⁿ (denoted CLB_H(ℝⁿ)) equipped with the finite‑valued Hausdorff metric, and the hyperspace of all non‑empty closed subsets of the real line ℝ (denoted CL_H(ℝ)) equipped with the extended (possibly infinite‑valued) Hausdorff metric.

First, the authors define CLB_H(X) = {A ⊆ X | A is closed and bounded} for a metric space (X,d) and endow it with the Hausdorff distance
d_H(A,B) = max{ sup_{a∈A} inf_{b∈B} d(a,b), sup_{b∈B} inf_{a∈A} d(a,b) }.
When X is complete, CLB_H(X) is also a complete metric space. The paper concentrates on X = ℝⁿ (n ≥ 1). Within CLB_H(ℝⁿ) the authors identify four natural dense subspaces:

1. Nowhere‑dense closed sets: N₁ = {A ∈ CLB_H(ℝⁿ) | int(A) = ∅}.
2. Perfect sets: N₂ = {A ∈ CLB_H(ℝⁿ) | A = A′, i.e., every point is an accumulation point}.
3. Cantor sets: N₃ = {A ∈ N₂ | A is compact, totally disconnected, zero‑dimensional}.
4. Lebesgue‑measure‑zero closed sets: N₄ = {A ∈ CLB_H(ℝⁿ) | λⁿ(A) = 0}.

Each N_i is shown to be dense in CLB_H(ℝⁿ) by explicit ε‑approximations: given any bounded closed set B and ε > 0, one can modify B slightly (e.g., remove a small open ball, add a thin dust‑like set, or intersect with a null‑measure Cantor set) to obtain a set in N_i within Hausdorff distance ε.

The core technical achievement is proving that every N_i is an absolute retract (AR) and has a countable base. The authors construct, for any finite‑dimensional polyhedron K and any continuous map f: K → N_i, a homotopy F: K ×