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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Paracompact product spaces defined by ultrafilters over the index set


Author: L. Brian Lawrence
Journal: Proc. Amer. Math. Soc. 108 (1990), 513-519
MSC: Primary 54D18; Secondary 54B10, 54B15, 54D40, 54G10
MathSciNet review: 987610
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Abstract: Let $ \omega = \{ 0,1, \ldots \} $, and suppose that for each $ i \in \omega ,{C_i}$ is a compact Hausdorff space with weight $ \leq c$. A filter over $ \omega $ defines a topology on $ \prod\nolimits_{i \in \omega } {{C_i}} $. We prove that the continuum hypothesis implies the existence of ultrafilters over $ \omega $ for which the corresponding product space on $ \prod\nolimits_{i \in \omega } {{C_i}} $ is paracompact. Moreover, we show that every $ {\mathbf{P}}$-point in $ \beta \omega - \omega $ is an ultrafilter with this property. Since box products appear as closed subspaces of ultrafilter products, our theorem extends results of Mary Ellen Rudin (1972) and Kenneth Kunen (1978).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0987610-1
PII: S 0002-9939(1990)0987610-1
Keywords: Paracompactness, box product, Stone-Čech remainder, $ {\mathbf{P}}$-point, $ {\mathbf{P}}$-space
Article copyright: © Copyright 1990 American Mathematical Society