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Proceedings of the American Mathematical Society

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

 

 

Complete bases and Wallman realcompactifications


Author: Jose L. Blasco
Journal: Proc. Amer. Math. Soc. 75 (1979), 114-118
MSC: Primary 54D60; Secondary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1979-0529226-2
MathSciNet review: 529226
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Abstract: We study a particular class of separating nest generated intersection rings on a Tychonoff space X, that we call complete bases. They are characterized by the equality $ \beta (\upsilon (X,\mathcal{D})) = \omega (X,\mathcal{D})$ between their associated Wallman spaces. It is proven that for each separating nest generated intersection ring $ \mathcal{D}$ there exists a unique complete base $ \hat{\mathcal{D}}$ such that $ \upsilon (X,\mathcal{D}) = \upsilon (X,\widehat{\mathcal{D}})$. From this result we obtain a necessary and sufficient condition for the existence of a continuous extension to $ \upsilon (X,\mathcal{D})$ of a real-valued function over X. Some applications of these results to certain inverse-closed subalgebras of $ C(X)$ are given.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0529226-2
Keywords: Nest generated intersection ring, strong delta normal base, complete base, countable intersection property, Q-closure, Q-dense, algebra, $ \sigma $-algebra
Article copyright: © Copyright 1979 American Mathematical Society