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

Taras Banakh, and Robert Cauty. "On universality of countable and weak products of sigma hereditarily disconnected spaces." Fundamenta Mathematicae 167.2 (2001): 97-109. <http://eudml.org/doc/282032>.

@article{TarasBanakh2001,
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 $X^\{ω\}$ 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.},
author = {Taras Banakh, Robert Cauty},
journal = {Fundamenta Mathematicae},
keywords = {universality; countable product; weak product; sigma hereditarily disconnected space; Nagata universal space; Polish space; Smirnov space; -space; -space; -space},
language = {eng},
number = {2},
pages = {97-109},
title = {On universality of countable and weak products of sigma hereditarily disconnected spaces},
url = {http://eudml.org/doc/282032},
volume = {167},
year = {2001},
}

TY - JOUR
AU - Taras Banakh
AU - Robert Cauty
TI - On universality of countable and weak products of sigma hereditarily disconnected spaces
JO - Fundamenta Mathematicae
PY - 2001
VL - 167
IS - 2
SP - 97
EP - 109
AB - 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.
LA - eng
KW - universality; countable product; weak product; sigma hereditarily disconnected space; Nagata universal space; Polish space; Smirnov space; -space; -space; -space
UR - http://eudml.org/doc/282032
ER -