On universality of countable and weak products of sigma hereditarily disconnected spaces
Spaces considered in this work are separable and metrizable. If \(\mathcal C\) is a class of spaces and \(X\) is a space, then one refers to \(X\) as being universal for \(\mathcal C\) if each \(C\in\mathcal C\) admits a closed embedding in \(X\). A space \(X\) is said to be sigma hereditarily disconnected if it is a countable union of hereditarily disconnected spaces. Recall also that a Polish space is one which is completely metrizable. NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINEHere are examples of some of the results obtained in this paper: NEWLINENEWLINENEWLINE(1) Let \(Y\) be sigma hereditarily disconnected and \(X\subset Y\). Then the countable power \(X^\omega\) is not universal for the class \(\mathcal A_2\) of absolute \(G_{\delta\sigma}\)-sets. NEWLINENEWLINENEWLINE(2) The countable power \(X^\omega\) of any absolute retract of the first Baire category is universal for the class of all \(\sigma\)-compact spaces having a strongly countable-dimensional completion. NEWLINENEWLINENEWLINE(3) Let \(X\) be a Polish space and \(Y\subset X\) admit an embedding into a \(\sigma\)-compact hereditarily disconnected space \(Z\). Then the weak product \(W(X,Y)=\{(x_i)\in X^\omega:\text{almost all }x_i\in Y\}\subset X^\omega\) is not universal for the class \(\mathcal M_3\) of absolute \(G_{\delta\sigma\delta}\)-sets.
scientific article; zbMATH DE number 1605239