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Separating by G δ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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.

Some cardinal characteristics of ordered sets

Vítězslav Novák (1998)

Czechoslovak Mathematical Journal

For ordered (= partially ordered) sets we introduce certain cardinal characteristics of them (some of those are known). We show that these characteristics—with one exception—coincide.

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