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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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References:
 [1] Balcar, B.; Hernández, F.; Hrušák, M., Combinatorics of dense subsets of the rationals, Fund. math., 183, 59-80, (2004) · Zbl 1051.03038 [2] Balcar, B.; Pelant, J.; Simon, P., The space of ultrafilters on N covered by nowhere dense sets, Fund. math., 110, 11-24, (1980) · Zbl 0568.54004 [3] Balcar, B.; Simon, P., Disjoint refinement, (), 333-386 [4] Bartoszyński, T.; Judah, H., Set theory, on the structure of the real line, (1995), A.K. Peters · Zbl 0834.04001 [5] Dordal, P.L., A model in which the base-matrix tree can not have cofinal branches, J. symbolic logic, 52, 651-664, (1987) · Zbl 0637.03049 [6] van Douwen, E., Transfer of information about $$\beta \mathbb{N} \smallsetminus \mathbb{N}$$ via open remainder maps, Illinois J. math., 34, 769-792, (1990) · Zbl 0709.54020 [7] Dow, A., The regular open algebra of $$\beta \mathbb{R} \smallsetminus \mathbb{R}$$ is not equal to the completion of $$\mathcal{P}(\omega) / \operatorname{fin}$$, Fund. math., 157, 33-41, (1998) [8] Dow, A., Tree π-bases for βN∖N in various models, Topology appl., 33, 3-19, (1989) · Zbl 0697.54003 [9] Farah, I., OCA and towers in $$\mathcal{P}(\omega) / \operatorname{fin}$$, Comment. math. univ. carolin., 37, 861-866, (1996) · Zbl 0887.03037 [10] Jech, T., Set theory, (2003), Springer Berlin · Zbl 1007.03002 [11] Keremedis, K., On the covering and the additivity number of the real line, Proc. amer. math. soc., 123, 1583-1590, (1995) · Zbl 0823.03026 [12] Koppelberg, S., () [13] Kunen, K., Set theory. an introduction to independence proofs, (1980), North-Holland Amsterdam · Zbl 0443.03021
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