Abstract
We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality \(\aleph _1\) the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection properties.
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10 October 2022
A Correction to this paper has been published: https://doi.org/10.1007/s13398-022-01331-4
Notes
It was pointed out by the Referee that the proof of Proposition 2.19 is valid without the assumption on the regularity of \(\kappa \).
References
Abraham, U., Bonnet, R., Kubiś, W., Rubin, M.: On Poset Boolean algebras. Order 20, 265–290 (2003)
Alster, K.: Some remarks on Eberlein compacts. Fund. Math. 104, 43–46 (1979)
Balcar, B., Franek, F.: Independent families in complete Boolean algebras. Trans. Am. Math. Soc. 274, 607–618 (1982)
Bandlow, I.: A construction in set-theoretic topology by means of elementary substructures. Z. Math. Logik Grundlag. Math. 37(5), 467–480 (1991)
Bandlow, I.: A characterization of Corson-compact spaces. Comment. Math. Univ. Carolin. 32, 545–550 (1994)
Bandlow, I.: On function spaces of Corson-compact spaces. Comment. Math. Univ. Carolin. 32, 347–356 (1994)
Bandelt, H.J., Hedlíková, J.: Median algebras. Discret. Math. 45, 1–30 (1983)
Bell, M.: The hyperspace of a compact space, I. Topol. Appl. 72, 39–46 (1996)
Bell, M., Marciszewski, W.: Function spaces on \(\tau \)-Corson compacta and tightness of polyadic spaces. Czechoslovak Math. J. 54(129), 899–914 (2004)
Bishop, E., de Leeuw, K.: The representations of linear functionals by measures on sets of extreme points. Ann. Inst. Fourier 9, 305–331 (1959)
Bonnet, R., Rubin, M.: On well-generated Boolean algebras. Ann. Pure Appl. Logic 105, 1–50 (2000)
Dow, A.: An introduction to applications of elementary submodels to topology. Topol. Proc. 13, 17–72 (1988)
Gruenhage, G.: Covering properties on \(X^2 \setminus \Delta \), \(W\)-sets, and compact subsets of \(\Sigma \)-products. Topol. Appl. 17, 287–304 (1984)
Gul’ko, S.P.: Properties of sets that lie in \(\Sigma \)-products (Russian). Dokl. Akad. Nauk SSSR 237, 505–508 (1977)
Hofmann, K.H., Mislove, M., Stralka, A.: Pontryagin duality of compact \(0\)-dimensional semilattices and its applications. Lect. Notes Math. 396, 122 (1991)
Juhàsz, I.: Cardinal Functions in Topology—Ten Years Later, Mathematisch centrum Amsterdam, p 168 (1983)
Just, W., Weese, M.: Discovering Modern Set Theory. II. Set-Theoretic Tools for Every Mathematician. Graduate Studies in Mathematics, vol. 18, p. xiv+224. American Mathematical Society, Providence (1997)
Ka̧kol J., Kubiś W., López-Pellicer M.: Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics, vol. 24, p. xii+493. Springer, Berlin (2011)
Kalenda, O.: Valdivia compact spaces in topology and Banach space theory. Extracta Math. 15, 1–85 (2000)
Kalenda, O.: Natural examples of Valdivia compact spaces. J. Math. Anal. Appl 350, 464–484 (2009)
Koppelberg, S.: In: Monk, J.D. (ed.) Handbook on Boolean Algebras, vol. 1, p. xx+312. North-Holland Publishing, Amsterdam (1989)
Kubiś, W., Moltó, A., Troyanski, S.: Topological properties of the continuous function spaces on some ordered compacta. Set-Valued Var. Anal. 21, 649–659 (2013)
Kunen, K.: Set Theory, Studies in Logic, vol. 102, p. xvi+313. North Holland, Amsterdam (1980)
Michael, E., Rudin, M.E.: A note on Eberlein compacts. Pac. J. Math. 72, 487–495 (1977)
Monk J.D.: Cardinal Invariants on Boolean Algebras, Third edition, Progress in Mathematics, Birkhäuser Verlag, p. 298. http://euclid.colorado.edu/~monkd/monk63.pdf (1996)
Monk, J.D.: Maximal free sequences in a Boolean algebra. Comment. Math. Univ. Carolin. 52(4), 593–610 (2011)
Plebanek G.: On \(\kappa \)-Corson Compacta. arXiv:2107.02513 (preprint)
Rosenstein, J.G.: Linear Orderings, p. 487. Academic Press, New York (1982)
Todorčević, S.: The functor \( ^2X\). Stud. Math. 116, 49–57 (1995)
Todorčević, S.: Trees and linearly ordered sets. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, p. vii + 1273. North Holland, Amsterdam (1984)
van de Vel, M.L.J.: Theory of Convex Structures, North-Holland Mathematical Library, vol. 50, p. 539. North Holland, Amsterdam (1993)
Acknowledgements
We would like to thank Grzegorz Plebanek and Witold Marciszewski for very useful comments and suggestions, that lead to a significant improvement of the presentation.
Funding
The third author was partially supported by grants from NSERC (455916), CNRS (UMR7586) and SFRS(7750027).
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R. Bonnet was supported by the Institute of Mathematics of the Czech Academy of Sciences, Prague.
W. Kubiś was supported by the GA ČR Grant 20-22230L (Czech Science Foundation)
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Bonnet, R., Kubiś, W. & Todorčević, S. Ultrafilter selection and Corson compacta. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 178 (2022). https://doi.org/10.1007/s13398-022-01317-2
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DOI: https://doi.org/10.1007/s13398-022-01317-2