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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[{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\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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DOI: https://doi.org/10.1090/S0002-9947-1982-0642334-5
Article copyright: © Copyright 1982 American Mathematical Society