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Transactions of the American Mathematical Society

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

 
 

 

An internal characterization of paracompact $ p$-spaces


Author: R. A. Stoltenberg
Journal: Trans. Amer. Math. Soc. 196 (1974), 249-263
MSC: Primary 54D20
DOI: https://doi.org/10.1090/S0002-9947-1974-0346746-4
MathSciNet review: 0346746
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Abstract: The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences $ \bmod\;k$. A space X has a refining sequence $ \bmod\;k$ if there exists a sequence $ \{ {\mathcal{G}_n}\vert n \in N\} $ of open covers for X such that $ \cap _{n = 1}^\infty {\text{St}}(C,{\mathcal{G}_n}) = P_C^1$ is compact for each compact subset C of X and $ {\text{\{ St}}{(C,{\mathcal{G}_n})^ - }\vert n \in N\} $ is a neighborhood base for $ P_C^1$. If $ P_C^1 = C$ for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if $ P_C^1 = C$ for all such sets then X is developable. Thus the concept of a refining sequence $ \bmod\;k$ is natural and it is helpful in understanding paracompact p-spaces.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0346746-4
Keywords: Metric spaces, paracompact p-spaces, developable spaces, sequences of open covers, subparacompact spaces
Article copyright: © Copyright 1974 American Mathematical Society

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