Stone spaces and compactifications (Q2929803)

The authors study clopen (closed and open) sets in compactifications. In particular, they study those spaces whose compactification is a Stone space (zero dimensional \(T_0\) compact space).NEWLINENEWLINECharacterizations are given for the Alexandroff, Wallman and Stone-Čech compactifications.NEWLINENEWLINESo, in Proposition 3.1 it is proved that the Alexandroff compactification is a Stone space if and only if the space is a Hausdorff space and the collection of compact clopen sets is a base for the topology. In Proposition 3.3 it is proved that for a \(T_1\) space the Wallman compactification is a Stone space if and only if disjoint closed sets can be separated by a clopen set. Finally, a similar result is proven for the Stone-Čech compactification.