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Transactions of the American Mathematical Society

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

 
 

 

Calibers of compact spaces


Authors: S. Argyros and A. Tsarpalias
Journal: Trans. Amer. Math. Soc. 270 (1982), 149-162
MSC: Primary 54A25; Secondary 06E10
DOI: https://doi.org/10.1090/S0002-9947-1982-0642334-5
MathSciNet review: 642334
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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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Article copyright: © Copyright 1982 American Mathematical Society