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

Fundamenta Mathematicae (2001)

- Volume: 167, Issue: 2, page 97-109
- ISSN: 0016-2736

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topTaras 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 -

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