Calibers of compact spaces

Abstract:

Let $X$ be a compact Hausdorff space and $\kappa$ its Souslin number.$^{2}$ We prove that if $\alpha$ is a cardinal such that either $\alpha$ and $\operatorname {cf} (\alpha )$ are greater than $\kappa$ and strongly $\kappa$-inaccessible or else $\alpha$ is regular and greater than $\kappa$, then $X$ has $(\alpha , \sqrt [\underparen {\kappa }]{\alpha })$ caliber. Restricting our interest to the category of compact spaces $X$ with $S(X) = {\omega ^ + }$ (i.e. $X$ satisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact space $X$ with $S(X) = {\omega ^ + }$, we have that (a) if $\alpha$ is a cardinal and $\operatorname {cf} (\alpha )$ does not have the form ${\beta ^ + }$ with $\operatorname {cf} (\beta ) = \omega$, then $\alpha$ is caliber for the space $X$. (b) If $\varepsilon = {\beta ^ + }$ and $\operatorname {cf} (\beta ) = \omega$ then $(\alpha , \beta )$ is caliber for $X$. A related example shows that the result of (b) is in a sense the best possible.