The Stone-Cech compactifications of ${\omega }^{\ast }\setminus \{x\}$ and $S_{\kappa }\setminus \{x\}$

Abstract

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 ${\omega }^{\ast }$. 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 ${\omega }^{\ast }$. 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 ${\omega }^{\ast }\setminus \{x\}$ are all homeomorphic.

18 pages. V3: Extended version; Thm 6.3 proves an open conjecture from V2