Frolík’s theorem for basically disconnected spaces. (English) Zbl 0799.54030

Vermeer, J.

Acta Univ. Carol., Math. Phys. 34, No. 2, 135-142 (1993).

A space \(X\) is called \(\kappa\)-basically disconnected if it is the Stone space of a \(\kappa\)-complete Boolean algebra. The main idea of the paper under review is to obtain Frolík’s theorem as a “limit case” of some theorem on self-embeddings of \(\kappa\)-basically disconnected compact spaces. One of the theorems says that if \(F\) is the fixed-point set of an embedding of a \(\kappa\)-basically disconnected space \(X\), then \(F\) is a \(P_ \kappa\)-set, i.e. for every family \({\mathcal R}\) of less than \(\kappa\) open neighbourhoods of \(F\) there exists an open set \(U\) such that \(F \subset U \subset \bigcap {\mathcal R}\). The paper contains also a nice extension of yet another theorem of Frolík: for every autohomeomorphism \(f\) of a basically disconnected compact space \(X\) every fixed point has a clopen base which is both \(f\)-invariant and \(f^{-1}\)- invariant. Extending a result of Abramovich, Arenson and Kitover [Report Berkeley, no. MSR 1 05808-91] the author proved that if \(f\) is an arbitrary continuous self-map of a basically disconnected compact space and \(F^ \#\) is the smallest closed \(f^{-1}\)-invariant set containing the set \(F\) of all fixed points, then \(F^ \#\) is a \(P\)- set.

Reviewer: A.Błaszczyk (Katowice)

MSC:

54G05	Extremally disconnected spaces, \(F\)-spaces, etc.
06E15	Stone spaces (Boolean spaces) and related structures

Keywords:

basically disconnected space; \(P\)-set