Interpolation of $\kappa$-compactness and PCF

Juhász, István ; Szentmiklóssy, Zoltán

Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 2, pp. 315-320.

Résumé

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 $\kappa$.

Détail

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