It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-1-5,
author = {Yasushi Hirata and Nobuyuki Kemoto},
title = {Separating by $G\_{d}$-sets in finite powers of o1},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {83-94},
zbl = {1029.54026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-1-5}
}
Yasushi Hirata; Nobuyuki Kemoto. Separating by $G_{δ}$-sets in finite powers of ω₁. Fundamenta Mathematicae, Tome 177 (2003) pp. 83-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-1-5/