# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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by S. Argyros and A. Tsarpalias
Trans. Amer. Math. Soc. 270 (1982), 149-162 Request permission

## 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.
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