Zoltán Szentmiklóssy
Interpolation of $\kappa$-compactness and PCF

Comment.Math.Univ.Carolinae 50,2 (2009) 315-320.

Abstract: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~ $\kappa$.

Keywords: complete accumulation point, $\kappa$-compact space, linearly Lindelöf spa\-ce, PCF theory
AMS Subject Classification: 03E04 54A25 54D30

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