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

Abstract

Suppose a metrizable separable space Y is sigma hereditarily discon- nected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ! of any subspace X Y is not universal for the class A2 of absolute G -sets; moreover, if Y is an absolute F -set, then X ! contains no closed topological copy of the Nagata spaceN = W (I;P); if Y is an absolute G -set, then X ! contains no closed copy of the Smirnov space = W (I; 0). On the other hand, the countable power X ! of any absolute retract of the rst Baire category contains a closed topological copy of each -compact space having a strongly countable-dimensional completion. We also prove that for a Polish space X and a subspace Y X admitting an em- bedding into a -compact sigma hereditarily disconnected space Z the weak product W (X;Y ) =f(xi)2 X ! : almost all xi 2 Yg X ! is not universal for the classM3 of absolute G -sets; moreover, if the space Z is compact then W (X;Y ) is not universal for the classM2 of absolute F -sets.