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

Taras Banakh ¹   ; Robert Cauty ²

¹ Department of Mathematics Lviv University Universytetska 1 Lviv 79000, Ukraine
² Université Paris VI UFR 920 Boîte courrier 172 4, Place Jussieu 75252 Paris Cedex 05, France

Fundamenta Mathematicae, Tome 167 (2001) no. 2, pp. 97-109

Résumé

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.

Détail

DOI : 10.4064/fm167-2-1

Keywords: suppose metrizable separable space sigma hereditarily disconnected countable union hereditarily disconnected subspaces prove countable power omega subspace subset universal class cal absolute delta sigma sets moreover absolute sigma delta set omega contains closed topological copy nagata space cal mathbb absolute delta set omega contains closed copy smirnov space sigma other countable power omega absolute retract first baire category contains closed topological copy each sigma compact space having strongly countable dimensional completion prove polish space subspace subset admitting embedding sigma compact sigma hereditarily disconnected space weak product omega almost subset omega universal class cal absolute delta sigma delta sets moreover space compact universal class cal absolute sigma delta sets