On universality of countable and weak products of sigma hereditarily disconnected spaces

Abstract

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces We prove that the countable power Xω of any subspace X ⊂ Y is not universal for the class A2 of absolute Gδσ-sets; moreover, if Y is an absolute Fσδ-set, then Xω contains no closed topological copy of the Nagata space N = W (I, ℙ); if Y is an absolute Gδ-set, then Xω contains no closed copy of the Smirnov space σ = W (I, 0). On the other hand, the countable power Xω of any absolute retract of the first Baire category contains a closed topological copy of each σ-compact space having a strongly countable-dimensional completion. We also prove that for a Polish space X and a subspace Y ⊂ X admitting an embedding into a σ-compact sigma hereditarily disconnected space Z the weak product W (X, Y) = {(xi) ∈ Xω : almost all xi ∈ Y} ⊂ Xω is not universal for the class M3 of absolute Gδσδ-sets-, moreover, if the space Z is compact then W (X, Y) is not universal for the class M2 of absolute Fσδ-sets.