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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
4
Pages
18-24
 Section Mathematics Title On the density and Suslin number of subsets of one Stone space Author(-s) Gryzlov A.A.a, Golovastov R.A.a Affiliations Udmurt State Universitya Abstract The paper concerns the Stone space of the Boolean algebra of subsets of one countable partially ordered set. The main feature of this set is the existence of countably many successors of each of its elements. From this property it follows that every fixed ultrafilter of this Stone space is a nonisolated point; the subset of free ultrafilters is dense everywhere. The classification of space points is given; the fact that there are free ultrafilters, which are not limits of sequences of fixed ultrafilters, as well as free ultrafilters determined by chains of partially ordered set, is proved. The cardinal invariants of the subspace of free ultrafilters are considered. It is shown that this subspace has the countable Suslin number, but is not separable. Keywords Boolean algebra, Stone space of Boolean algebra, ultrafilter UDC 515.122.536 MSC 54D35 DOI 10.20537/vm140402 Received 10 October 2014 Language Russian Citation Gryzlov A.A., Golovastov R.A. On the density and Suslin number of subsets of one Stone space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 18-24. References Bell M.G. Compact ccc non-separable spaces of small weight, Topology Proceedings, 1980, vol. 5, pp. 11-25. Golovastov R.A. About one compactifications of countable discrete space, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 14-19 (in Russian). Golovastov R.A. About Stone space of one Boolean algebra, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 3, pp. 19-24 (in Russian). Gryzlov A.A. On compactifications of discrete spaces, Fundam. Prikl. Mat., 1996, vol. 2, no. 3, pp. 803-848 (in Russian). Gryzlov A.A., Bastrykov E.S., Golovastov R.A. On Bell's compactification of $N$, Topology Proceedings, 2010, vol. 35, pp. 177-185. Gryzlov A.A., Bastrykov E.S., Golovastov R.A. About points of compactification of $N$, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 10-17 (in Russian). Gryzlov A.A., Bastrykov E.S. Some centered systems of sets and points defined by them, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 4, pp. 76-82 (in Russian). Gryzlov A.A., Golovastov R.A. The Stone spaces of Boolean algebras and the Cantor set, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 1, pp. 11-16 (in Russian). Gryzlov A.A. On convergent sequence and copies of $\beta{N}$ in the Stone space of one Boolean algebra, Topology Proceedings, 2013, vol. 42, pp. 165-171. van Mill J. Weak $p$-points in Chech-Stone compactification, Trans. Amer. Math. Soc., 1980, vol. 173, no. 2, pp. 657-678. Full text