Geodesic
Interpolation of $\kappa$-compactness and PCF
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 2, pp. 315-320 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\kappa$.
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 $\kappa$.
Classification : 03E04, 54A25, 54D30
Keywords: complete accumulation point; $\kappa$-compact space; linearly Lindelöf space; PCF theory 
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     title = {Interpolation of $\kappa$-compactness and {PCF}},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {315--320},
     year = {2009},
     volume = {50},
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     zbl = {1212.03029},
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Juhász, István; Szentmiklóssy, Zoltán. Interpolation of $\kappa$-compactness and PCF. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 2, pp. 315-320. http://geodesic.mathdoc.fr/item/CMUC_2009_50_2_a10/