Exponents of products of metric spaces (Q1107864)

The main result of the paper is that for every collection of metric (resp. 0-dimensional metric) spaces \(\{M_{\alpha}\}_{\alpha <\omega_ 1}\) there exists a collection \(\{M'_{\alpha}\}_{\alpha <\omega_ 1}\) of metric (resp. 0-dimensional metric) spaces and a perfect map (resp. perfect retraction) f: \(\prod_{\alpha <\omega_ 1}M'_{\alpha}\to \exp (\prod_{\alpha <\omega_ 1}M_{\alpha})\). (For a topological space X, exp X denotes the space of all nonempty compact subsets of X, equipped with the Vietoris topology.) In a certain sense, this result generalizes the work of \textit{S. Sirota} [Dokl. Akad. Nauk SSSR 181, No.5, 1069-1072 (1968; Zbl 0184.264)] who proved that exp \(D^{\aleph_ 1}\) is homeomorphic to \(D^{\aleph_ 1}\) \(=\) the Cantor cube of weight \(\aleph_ 1\). The present theorem is, in a way, the best possible since \textit{L. B. Shapiro} [Dokl. Akad. Nauk SSSR 228, No.6, 1302-1305 (1976; Zbl 0342.54031)], who proved that exp \(D^{\aleph_ 2}\) isn't even a dyadic compactum (i.e. the image of a Cantor cube).