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

Yasushi Hirata, and Nobuyuki Kemoto. "Separating by $G_{δ}$-sets in finite powers of ω₁." Fundamenta Mathematicae 177.1 (2003): 83-94. <http://eudml.org/doc/282835>.

@article{YasushiHirata2003,
abstract = {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.},
author = {Yasushi Hirata, Nobuyuki Kemoto},
journal = {Fundamenta Mathematicae},
keywords = {subnormal; subshrinking; product; ordinal; stationary set; Pressing Down Lemma; -sets},
language = {eng},
number = {1},
pages = {83-94},
title = {Separating by $G_\{δ\}$-sets in finite powers of ω₁},
url = {http://eudml.org/doc/282835},
volume = {177},
year = {2003},
}

TY - JOUR
AU - Yasushi Hirata
AU - Nobuyuki Kemoto
TI - Separating by $G_{δ}$-sets in finite powers of ω₁
JO - Fundamenta Mathematicae
PY - 2003
VL - 177
IS - 1
SP - 83
EP - 94
AB - 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.
LA - eng
KW - subnormal; subshrinking; product; ordinal; stationary set; Pressing Down Lemma; -sets
UR - http://eudml.org/doc/282835
ER -