Lyubomyr Zdomskyy
A semifilter approach to selection principles II: $\tau ^\ast $-covers

Comment.Math.Univ.Carolin. 47,3 (2006) 539-547.

Abstract:Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property $\bigcup _{\text {\rm fin}}(\Cal O, \text {\rm T}^\ast )$ provided $(\frak u<\frak g)$, and every space with the property $\bigcup _{\text {\rm fin}}(\Cal O, \text {\rm T}^\ast )$ is Hurewicz provided $(\text {\rm Depth}^+([\omega ]^{\aleph _0})\leq \frak b)$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text {\mathsf P}$ and $\text {\mathsf Q}$ [do not] coincide, where $\text {\mathsf P}$ and $\text {\mathsf Q}$ run over $\bigcup _{\text {\rm fin}}(\Cal O,\Gamma )$, $\bigcup _{\text {\rm fin}}(\Cal O, \text {\rm T})$, $\bigcup _{\text {\rm fin}}(\Cal O, \text {\rm T}^\ast )$, $\bigcup _{\text {\rm fin}}(\Cal O, \Omega )$, and $\bigcup _{\text {\rm fin}}(\Cal O, \Cal O)$.

Keywords: selection principle, semifilter, small cardinals
AMS Subject Classification: 03A, 03E17, 03E35