Separating by $G_{\delta }$-sets in finite powers of $\omega _1$
Volume 177 / 2003
Fundamenta Mathematicae 177 (2003), 83-94
MSC: 54B10, 03E10, 54D15, 54D20.
DOI: 10.4064/fm177-1-5
Abstract
It is known that all subspaces of $\omega _1^2$ have the property that every pair of disjoint closed sets can be separated by disjoint $G_{\delta } $-sets (see [4]). It has been conjectured that all subspaces of $\omega _1^n$ also have this property for each $n<\omega $. We exhibit a subspace of $\{ \langle \alpha ,\beta ,\gamma \rangle \in \omega _1^3:\alpha \leq \beta \leq \gamma \} $ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of $\{ \langle \alpha ,\beta ,\gamma \rangle \in \omega _1^3:\alpha <\beta <\gamma \} $ have this property.
