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A semifilter approach to selection principles. II: ▫$\tau^\ast$▫-covers
Zdomskyy, Lyubomyr, 1983-
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 ▫$\bigcup_{\text{fin}}(\mathcal{O}, ...\text{T}^\ast )$▫ provided ▫$(\mathfrak{u} < \mathfrak{g})$▫. Moreover we show that every space with the property ▫$\bigcup_{\text{fin}}(\mathcal{O}, \text{T}^\ast )$▫ is Hurewicz provided ▫$(\text{Depth}^+ ([\omega ]^{\aleph_0}) \leq \mathfrak{b})$▫. Combining this with the results proven in the cited literature, we settle all questions whether (it is consistent that) the properties ▫$\text{P}$▫ and ▫$\text{Q}$▫ [do not] coincide, where ▫$\text{P}$▫ and ▫$\text{Q}$▫ run over ▫$\bigcup_{\text{fin}} (\mathcal{O}, \Gamma)$▫, ▫$\bigcup_{\text{fin}} (\mathcal{O}, \text{T})$▫, ▫$\bigcup_{\text{fin}} (\mathcal{O}, \text{T}^\ast)$▫, ▫$\bigcup_{\text{fin}} (\mathcal{O}, \Omega)$▫, and ▫$\bigcup_{\text{fin}} (\mathcal{O}, \mathcal{O})$▫.
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