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Separating by $G_{δ}$-sets in finite powers of ω₁

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Hirata Y., Kemoto N.

Fundamenta Mathematicae

2003

tom 177

nr 1

83-94

It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{δ}$-sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of {⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ} which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of {⟨α,β,γ⟩ ∈ ω₁³: α < β < γ} have this property.

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1 Fundamenta Mathematicae

1 Hirata Y.

1 Kemoto N.

1 2003

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Separating by $G_{δ}$-sets in finite powers of ω₁