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

• Autorzy: Taras Banakh , Robert Cauty
• Języki publikacji: EN

Abstrakty

EN

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 𝓐₂ 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 𝓝 = 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.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 54B10: Product spaces
• 55N10: Singular theory
• 55M10: Dimension theory
• 57N20: Topology of infinite-dimensional manifolds
• 54F45: Dimension theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 167
• Numer: 2
• Strony: 97-109

Daty

wydano

2001

Twórcy

autor

Taras Banakh

• Department of Mathematics, Lviv University, Universytetska 1, Lviv 79000, Ukraine

autor

Robert Cauty

• Université Paris VI, UFR 920, Boîte courrier 172, 4, Place Jussieu, 75252 Paris Cedex 05, France

Identyfikatory

DOI

10.4064/fm167-2-1