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

Taras Banakh ; Robert Cauty

Fundamenta Mathematicae, Tome 167 (2001), p. 97-109 / Harvested from The Polish Digital Mathematics Library

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 $X^{ω}$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{δσ}$-sets; moreover, if Y is an absolute $F_{σδ}$-set, then $X^{ω}$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{δ}$-set, then $X^{ω}$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power $X^{ω}$ 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 $W(X,Y) = {(x_i) ∈ X^{ω}: almost all x_i ∈ Y} ⊂ X^{ω}$ is not universal for the class ℳ ₃ of absolute $G_{δσδ}$-sets; moreover, if the space Z is compact then W(X,Y) is not universal for the class ℳ ₂ of absolute $F_{σδ}$-sets.

Détail

Publié le : 2001-01-01

Zbl 0973.54017

EUDML-ID : urn:eudml:doc:282032