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

Max F. Pitz, Rolf Suabedissen

Published 2013-10-02, updated 2014-05-30Version 3

The space $S_\kappa$ is the Stone space of the $\kappa$-saturated Boolean algebra of cardinality $\kappa$. It exists provided that $\kappa = \kappa^{<\kappa}$, and is characterised topologically as the unique $\kappa$-Parovichenko space of weight $\kappa$. Under the Continuum Hypothesis, $S_{\omega_1}$ coincides with $\omega^*$. This paper investigates questions related to the Stone-Cech compactification of spaces $S_\kappa \setminus \{x\}$, extending corresponding results obtained by Fine & Gillman and Comfort & Negrepontis for the space $\omega^*$. We show that for every point $x$ of $S_\kappa$, the Stone-Cech remainder of $S_\kappa \setminus \{x\}$ is a $\kappa^+$-Parovichenko space of cardinality $2^{2^\kappa}$ which admits a family of $2^\kappa$ disjoint clopen sets. As a corollary we get that it is consistent with CH that the Stone-Cech remainders of $\omega^* \setminus \{x\}$ are all homeomorphic.