Strongly countable dimensional compacta form the Hurewicz set
\textit{R.~Cauty} [Fundam. Math. 138, 35--58 (1991; Zbl 0770.54015)] proved that the metric space of all differentiable functions \(f:[0,1]\to\mathbb R\) with the supremum norm is homeomorphic to the Hurewicz set~\(\mathcal H\) of countable compact subsets of \([0,1]\). \textit{H.~Gladdines} and \textit{J. van Mill} [ibid. 142, 173--188 (1993; Zbl 0811.54028)] proved that the hyperspace of locally connected continua in the cube \([0,1]^3\) is homeomorphic to the infinite Cartesian power of the pseudoboundary of the Hilbert cube~\(\ell^\infty\). Let us recall that a~space is strongly countable dimensional if it is a~countable union of closed finite dimensional subsets. The authors of the paper under review apply some of the ideas of the above mentioned papers (in particular, the powerful method of absorbers) and prove that the hyperspace of all strongly countable dimensional compacta in~\(\ell^\infty\) of dimension at least~\(k\) with \(k\geq1\) and the hyperspace of all strongly countable dimensional continua in~\(\ell^\infty\) of dimension at least~\(k\) with \(k\geq2\) are homeomorphic to~\(\mathcal H\). These results hold for any true dimension in the sense of Dobrowolski and Rubin which includes the covering dimension dim and the cohomological dimension dim\(_G\) with respect to any Abelian group.
scientific article; zbMATH DE number 5131628