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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The compactifications to which an element of $ C\sp{\ast} (X)$ extends


Authors: Richard E. Chandler and Ralph Gellar
Journal: Proc. Amer. Math. Soc. 38 (1973), 637-639
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1973-0314003-2
MathSciNet review: 0314003
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Abstract: We first determine a necessary and sufficient condition for a function $ f \in {C^\ast}(X)$, which extends to a compactification of $ X$, to extend to a smaller compactification. We apply this result to show that when $ \vert\beta X\backslash X\vert \leqq {\aleph _0}$ there is an $ f \in {C^\ast}(X)$ which extends to no compactification other than $ \beta X$. Two examples show that when $ {\aleph _0} < \vert\beta X\backslash X\vert \leqq c$ no such definite result may be obtained.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0314003-2
Keywords: Hausdorff compactifications, Stone-Čech compactifications, extensions
Article copyright: © Copyright 1973 American Mathematical Society