The Stone-Cech compactifications of $ω^*\setminus \{x\}$ and $S_κ\setminus\{x\}$

Abstract

The space $S_κ$ is the Stone space of the $κ$-saturated Boolean algebra of cardinality $κ$. It exists provided that $κ= κ^{<κ}$, and is characterised topologically as the unique $κ$-Parovichenko space of weight $κ$. Under the Continuum Hypothesis, $S_{ω_1}$ coincides with $ω^*$. This paper investigates questions related to the Stone-Cech compactification of spaces $S_κ\setminus \{x\}$, extending corresponding results obtained by Fine & Gillman and Comfort & Negrepontis for the space $ω^*$. We show that for every point $x$ of $S_κ$, the Stone-Cech remainder of $S_κ\setminus \{x\}$ is a $κ^+$-Parovichenko space of cardinality $2^{2^κ}$ which admits a family of $2^κ$ disjoint clopen sets. As a corollary we get that it is consistent with CH that the Stone-Cech remainders of $ω^* \setminus \{x\}$ are all homeomorphic.