When is a cellular-countably-compact space, countably compact?
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On spaces with a $$\pi $$-base with elements with compact closure
In this paper we show that if X is a $$T_1$$ T 1 -space with a $$\pi $$ π -base whose elements have compact closure, then $$d(X)\le c(X)\cdot 2^{\psi (X)}$$ d ( X ) ≤ c ( X ) · 2 ψ ( X ) and therefore, for such spaces we have $$d(X)^{\psi (X)} = c(X)^{\psi (X)}$$ d ( X ) ψ ( X ) = c ( X ) ψ ( X ) . This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition, we show that for such spaces X Šapirovskiĭ’s inequality $$d(X)\le \pi \chi (X)^{c(X)}$$ d ( X ) ≤ π χ ( X ) c ( X ) , which is known to be true for regular Hausdorff spaces, is also valid. In the case when the space X is in addition sequential or radial, we show that $$|X|\le 2^{c(X)}$$ | X | ≤ 2 c ( X ) . This result extends two theorems of Arhangel$$'$$ ′ skiĭ to the class of Hausdorff spaces with a $$\pi $$ π -base whose elements have compact closures. We also show that spaces with a $$\pi $$ π -base with elements with compact closures are $$\alpha $$ α -favorable in the Banach–Mazur game, which implies such spaces are Baire. It was shown in Bella et al. (Quaest Math 46(4):745–760, 2023) that if a Hausdorff space X has a $$\pi $$ π -base consisting of elements with compact closure, then $$|X|\le 2^{wL(X)t(X)\psi _c(X)}$$ | X | ≤ 2 w L ( X ) t ( X ) ψ c ( X ) . We give a variation of this result by showing $$|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$$ | X | ≤ π χ ( X ) w L ( X ) ot ( X ) ψ c ( X ) for such a space X. Note that since $$wL(X)\textrm{ot}(X)\le c(X)$$ w L ( X ) ot ( X ) ≤ c ( X ) , this result is at least as good as that given by Sun (Proc Am Math Soc 104:313–316, 1988). We also give a possible improvement of the bound in Bella et al. (2023) by showing that $$|X|\le 2^{wL(X)wt(X)\psi _c(X)}$$ | X | ≤ 2 w L ( X ) w t ( X ) ψ c ( X ) for a Hausdorff space X with a $$\pi $$ π -base consisting of elements with compact closure. This uses the weak tightness wt(X) defined in Carlson (Topol Appl 249:103–111, 2018), which has the property $$\textrm{ot}(X)\le wt(X)\le t(X)$$ ot ( X ) ≤ w t ( X ) ≤ t ( X ) . We also show that if X is a Hausdorff homogeneous space with a $$\pi $$ π -base consisting of elements with compact closure (such spaces are locally compact), then $$|X|\le wL(X)^{wt(X)\pi \chi (X)}$$ | X | ≤ w L ( X ) w t ( X ) π χ ( X ) . This generalizes the result in Bella and Carlson (Monatsh Math 192(1):39–48, 2020) that if X is a homogeneous compactum, then $$|X|\le 2^{wt(X)\pi \chi (X)}$$ | X | ≤ 2 w t ( X ) π χ ( X ) .
On spaces with a $$\pi $$-base with elements with compact closure
In this paper we show that if X is a $$T_1$$ T 1 -space with a $$\pi $$ π -base whose elements have compact closure, then $$d(X)\le c(X)\cdot 2^{\psi (X)}$$ d ( X ) ≤ c ( X ) · 2 ψ ( X ) and therefore, for such spaces we have $$d(X)^{\psi (X)} = c(X)^{\psi (X)}$$ d ( X ) ψ ( X ) = c ( X ) ψ ( X ) . This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition, we show that for such spaces X Šapirovskiĭ’s inequality $$d(X)\le \pi \chi (X)^{c(X)}$$ d ( X ) ≤ π χ ( X ) c ( X ) , which is known to be true for regular Hausdorff spaces, is also valid. In the case when the space X is in addition sequential or radial, we show that $$|X|\le 2^{c(X)}$$ | X | ≤ 2 c ( X ) . This result extends two theorems of Arhangel$$'$$ ′ skiĭ to the class of Hausdorff spaces with a $$\pi $$ π -base whose elements have compact closures. We also show that spaces with a $$\pi $$ π -base with elements with compact closures are $$\alpha $$ α -favorable in the Banach–Mazur game, which implies such spaces are Baire. It was shown in Bella et al. (Quaest Math 46(4):745–760, 2023) that if a Hausdorff space X has a $$\pi $$ π -base consisting of elements with compact closure, then $$|X|\le 2^{wL(X)t(X)\psi _c(X)}$$ | X | ≤ 2 w L ( X ) t ( X ) ψ c ( X ) . We give a variation of this result by showing $$|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$$ | X | ≤ π χ ( X ) w L ( X ) ot ( X ) ψ c ( X ) for such a space X. Note that since $$wL(X)\textrm{ot}(X)\le c(X)$$ w L ( X ) ot ( X ) ≤ c ( X ) , this result is at least as good as that given by Sun (Proc Am Math Soc 104:313–316, 1988). We also give a possible improvement of the bound in Bella et al. (2023) by showing that $$|X|\le 2^{wL(X)wt(X)\psi _c(X)}$$ | X | ≤ 2 w L ( X ) w t ( X ) ψ c ( X ) for a Hausdorff space X with a $$\pi $$ π -base consisting of elements with compact closure. This uses the weak tightness wt(X) defined in Carlson (Topol Appl 249:103–111, 2018), which has the property $$\textrm{ot}(X)\le wt(X)\le t(X)$$ ot ( X ) ≤ w t ( X ) ≤ t ( X ) . We also show that if X is a Hausdorff homogeneous space with a $$\pi $$ π -base consisting of elements with compact closure (such spaces are locally compact), then $$|X|\le wL(X)^{wt(X)\pi \chi (X)}$$ | X | ≤ w L ( X ) w t ( X ) π χ ( X ) . This generalizes the result in Bella and Carlson (Monatsh Math 192(1):39–48, 2020) that if X is a homogeneous compactum, then $$|X|\le 2^{wt(X)\pi \chi (X)}$$ | X | ≤ 2 w t ( X ) π χ ( X ) .
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