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Proceedings of the American Mathematical Society

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

 

 

$ P$-sets in $ F\sp{'} $-spaces


Author: Robert E. Atalla
Journal: Proc. Amer. Math. Soc. 46 (1974), 125-132
MSC: Primary 54C05; Secondary 54G05
DOI: https://doi.org/10.1090/S0002-9939-1974-0348701-2
MathSciNet review: 0348701
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Abstract: A $ P$-set is one which is interior to any zero set which contains it. An $ F'$-space may be characterized as one in which the closure of a cozero set is a $ P$-set. We study applications of $ P$-sets to the topology of $ F'$-spaces, and certain set-theoretical operations under which the class of $ P$-sets is stable. A. I. Veksler has shown that in a basically disconnected space the closure of an arbitrary union of $ P$-sets is a $ P$-set, while in $ F'$-spaces we are only able to prove this for countable unions. Our main result is an example of a set in the compact $ F$-space $ \beta N\backslash N$ which is not a $ P$-set, but which is the closure of a union of $ P$-sets. The set is related to the almost-convergent functions of G. G. Lorentz.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0348701-2
Keywords: $ F$-space, basically disconnected, extremally disconnected, $ P$-set, countable chain condition, almost convergent sequence
Article copyright: © Copyright 1974 American Mathematical Society