On universality of countable and weak products of sigma hereditarily disconnected spaces
Volume 167 / 2001
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 $X^\omega $ of any subspace $X\subset Y$ is not universal for the class ${\cal A}_2$ of absolute $G_{\delta \sigma }$-sets; moreover, if $Y$ is an absolute $F_{\sigma \delta }$-set, then $X^\omega $ contains no closed topological copy of the Nagata space ${\cal N}=W(I,{\mathbb P})$; if $Y$ is an absolute $G_\delta $-set, then $X^\omega $ contains no closed copy of the Smirnov space ${\sigma }=W(I,0)$.
On the other hand, the countable power $X^\omega $ of any absolute retract of the first Baire category contains a closed topological copy of each ${\sigma }$-compact space having a strongly countable-dimensional completion.
We also prove that for a Polish space $X$ and a subspace $Y\subset X$ admitting an embedding into a ${\sigma }$-compact sigma hereditarily disconnected space $Z$ the weak product $W(X,Y)=\{ (x_i)\in X^\omega :$ almost all $x_i\in Y\} \subset X^\omega $ is not universal for the class ${\cal M}_3$ of absolute $G_{\delta {\sigma }\delta }$-sets; moreover, if the space $Z$ is compact then $W(X,Y)$ is not universal for the class ${\cal M}_2$ of absolute $F_{\sigma \delta }$-sets.
