Separating by $G_{\delta }$-sets in finite powers of $\omega _1$

Yasushi Hirata¹ ; Nobuyuki Kemoto²

¹ Graduate School of Mathematics University of Tsukuba Ibaraki 305-8571, Japan
² Department of Mathematics Faculty of Education Oita University Dannoharu, Oita, 870-1192, Japan

Fundamenta Mathematicae, Tome 177 (2003) no. 1, pp. 83-94

Résumé

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.

Détail

DOI : 10.4064/fm177-1-5

Keywords: known subspaces omega have property every pair disjoint closed sets separated disjoint delta sets see has conjectured subspaces omega have property each omega exhibit subspace langle alpha beta gamma rangle omega alpha leq beta leq gamma which does have property disproving conjecture other prove subspaces langle alpha beta gamma rangle omega alpha beta gamma have property