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Keywords:
selection principle; semifilter; small cardinals
selection principle; semifilter; small cardinals
Summary:
\font\mathsf=csss10 at 8pt 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 $\bigcup_{\operatorname{fin}}(\Cal O, \operatorname{T}^\ast)$ provided $(\frak u<\frak g)$, and every space with the property $\bigcup_{\operatorname{fin}}(\Cal O, \operatorname{T}^\ast)$ is Hurewicz provided $(\operatorname{Depth}^+([\omega]^{\aleph_0})\leq \frak b)$. 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 $\bigcup_{\operatorname{fin}}(\Cal O,\Gamma )$, $\bigcup_{\operatorname{fin}}(\Cal O, \operatorname{T})$, $\bigcup_{\operatorname{fin}}(\Cal O, \operatorname{T}^\ast)$, $\bigcup_{\operatorname{fin}}(\Cal O, \Omega )$, and $\bigcup_{\operatorname{fin}}(\Cal O, \Cal O)$.
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