Saturated Boolean algebras and their Stone spaces (Q1065415)

For \(\kappa\geq \omega\), a Tychonoff space X is called \(\kappa\)- Parovichenko if (1) X is compact and zero-dimensional, and (2) the Boolean algebra CO(X) of clopen subsets of X is \(\kappa\)-saturated and satisfies \(| CO(X)| =\kappa^{<\kappa}.\) It is a theorem of \textit{S. Negrepontis} [Trans. Am. Math. Soc. 141, 515-527 (1969; Zbl 0223.06002)] that for \(\kappa =\kappa^{<\kappa}\) there is a unique \(\kappa\)-Parovichenko space (denoted \(S_{\kappa})\). Every compact space of weight \(\leq \kappa^{<\kappa}\) is the continuous image of some \(\kappa\)-Parovichenko space, and every \(\kappa\)-Parovichenko space maps continuously onto every compact space of weight \(\leq \kappa.\) The present paper contains many theorems concerning spaces of Parovichenko type, some of them answering questions posed by Negrepontis (op. cit.). Among the more easily stated results are these. (2.3): The Stone-Čech remainder of the space of subuniform ultrafilters over \(\omega_ 1\) (that is, the remainder space \(\beta (SU(\omega_ 1))\setminus SU(\omega_ 1))\) is \(\omega_ 2\)-Parovichenko. (2.4): For \(\kappa\geq \omega\) the space of uniform ultrafilters over \(\kappa\) contains \(\kappa^+\)-Parovichenko spaces. (2.8): For \(\kappa >{\mathfrak c}\), every \(\kappa\)-Parovichenko space contains a compact extremally disconnected subspace of weight \(<\kappa\) which is not a retract. (3.2): For \(\kappa =\kappa^{<\kappa}\), a space X embeds as a nowhere dense \(P_{\kappa}\)-set in \(S_{\kappa}\) iff X is a compact \(F_{\kappa}\)- space with weight \(\leq \kappa\) (in particular, \(S_{\kappa}\) itself so embeds). 4.1: There is a unique \(\kappa\)-Parovichenko space iff \(\kappa =\kappa^{<\kappa}\).