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Distributivity of the algebra of regular open subsets of \(\beta\mathbb R\setminus \mathbb R\). (English) Zbl 1071.54018
The algebra \(RO(\beta\mathbb R\setminus\mathbb R)\) of regular open subsets of the Čech-Stone remainder of the real line is isomorphic to the completion of the Boolean algebra \(A^\omega/\text{Fin}\) where \(A\) is the Boolean algebra of clopen subsets of the Cantor set \(2^\omega\) and \(\text{Fin}\) is the set of elements of \(A^\omega\) with finite support. The authors prove that the distributivity number of the algebra \(A^\omega/\text{Fin}\) is below the distributivity number \(\mathfrak h\) of the algebra \(\mathcal P(\omega)/\text{fin}\) and below the additivity of the ideal of meager sets of reals and they prove that the tower number of \(A^\omega/\text{Fin}\) is equal to the tower number \(\mathfrak t\) of \(\mathcal P(\omega)/\text{fin}\). As a corollary they obtain simple arguments for a result of A. Dow who proved that in the iterated Mathias model the spaces \(\beta\omega\setminus\omega\) and \(\beta\mathbb R\setminus\mathbb R\) are not co-absolute. The authors also prove that under the assumption \(\mathfrak t=\mathfrak h\) the spaces \(\beta\omega\setminus\omega\) and \(\beta\mathbb R\setminus\mathbb R\) are co-absolute which improves the same result of E. van Douwen under the assumption \(\mathfrak p=\mathfrak h\).

MSC:
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
06E15 Stone spaces (Boolean spaces) and related structures
03E17 Cardinal characteristics of the continuum
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