𝑃-sets in 𝐹^{’}-spaces

Atalla, Robert E.

doi:10.1090/S0002-9939-1974-0348701-2

$P$-sets in $F^{’}$-spaces
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Proc. Amer. Math. Soc. 46 (1974), 125-132 Request permission

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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