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Keywords:
complete accumulation point; $\kappa$-compact space; linearly Lindelöf space; PCF theory
complete accumulation point; $\kappa$-compact space; linearly Lindelöf space; PCF theory
Summary:
We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi(\mu,\kappa,\lambda)$ denote the following statement: $\mu < \kappa < \lambda = \operatorname{cf} (\lambda)$ and there is $\{ S_\xi : \xi < \lambda \} \subset [\kappa]^\mu$ such that $|\{ \xi : |S_\xi \cap A| = \mu \}| < \lambda$ whenever $A \in [\kappa]^{<\kappa}$. We show that if $\Phi(\mu,\kappa,\lambda)$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi(\operatorname{cf} (\kappa), \kappa, \kappa^+)$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and $\aleph_\omega$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals $\kappa$.
References:
[1] Arhangel'skii A.V.: Homogeneity and complete accumulation points. Topology Proc. 32 (2008), 239--243. MR 1500085 | Zbl 1170.54009
[2] Shelah S.: Cardinal Arithmetic. Oxford Logic Guides, vol. 29, Oxford University Press, Oxford, 1994. MR 1318912 | Zbl 0864.03032
[3] van Douwen E.: The Integers and Topology. in Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds., North-Holland, Amsterdam, 1984, pp. 111--167. MR 0776619 | Zbl 0561.54004
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