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Proceedings of the American Mathematical Society

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

 

 

Regular matrices and $ P$-sets in $ \beta N\backslash N$


Author: R. E. Atalla
Journal: Proc. Amer. Math. Soc. 37 (1973), 157-162
MSC: Primary 54D40
MathSciNet review: 0324655
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Abstract: A P-set is a closed set which is interior to any zero set (closed $ {G_\delta }$) which contains it. Henriksen and Isbell showed that the 'support set' in $ \beta N\backslash N$ of a nonnegative regular matrix is a P-set. We show that each such support set contains a family of $ {2^c}$ pairwise disjoint perfect nowhere dense P-sets, so that not every P-set comes from a matrix. Moreover, each of the P-sets produced is the support of a Borel probability measure on $ \beta N\backslash N$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0324655-9
Keywords: Regular matrix, support set of a matrix, P-point, P-set, $ \beta N$
Article copyright: © Copyright 1973 American Mathematical Society