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Maximal nowhere dense P-sets in basically disconnected spaces and F-spaces. (English) Zbl 1053.54041
The authors investigate maximal nowhere dense sets in various types of compact spaces (a nowhere dense closed set \(\mathfrak M\) is maximal if there is no n.d. closed set \(\mathfrak M'\) containing \(\mathfrak M\) as a nowhere dense subset). As proved by P. Simon, the statement that there are no maximal n.d. sets in \(\omega ^*\) is equivalent to the Hechler conjecture.
The authors prove the existence of maximal n.d. sets which are moreover \(P\)-sets for Stone spaces of \(\omega _2\)-complete Boolean algebras. They have more results and characterization in this direction. The authors notice that there are many nowhere dense \(P\)-sets in a basically disconnected non-c.c.c. compact space but under [CH], there is no maximal n.d. \(P\)-set in \(\omega ^*\).
Reviewer: Jan Pelant (Praha)
MSC:
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54D30 Compactness
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