Separating by G_δ-sets in finite powers of ω₁

Abstract

Abstract.It is known that all subspaces of ω 2 1 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 ω n 1 also have this property for each n < ω. We exhibit a subspace of { α, β, γ ∈ ω 3 1 : 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 ∈ ω 3 1 : α < β < γ} have this property.

Mild normality of finite products of subspaces of ω1

“…Proofs of this lemma are seen in [1,2]. For reader's convenience, we give here a sketch of the proof.…”

“…On the other hand, A × B is mildly normal wherever A and B are arbitrary subsets of ordinals, see [3]. In [3], a subspace of ω 2 1 which is not mildly normal is given and they asked whether every finite product of subspaces of ordinals is mildly normal. On the other hand recently, it has been known that strong zero-dimensionality behaves like mild normality in the realm of products of ordinals.…”