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Ideal convergence of bounded sequences

Published online by Cambridge University Press:  12 March 2014

Rafał Filipów ,
Nikodem Mrożek ,
Ireneusz Recław  and
Piotr Szuca
Rafał Filipów
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, E-mail: [email protected], URL: http://www.math.univ.gda.pl/~rfilipow
Nikodem Mrożek
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, E-mail: [email protected]
Ireneusz Recław
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, E-mail: [email protected], URL: http://www.math.univ.gda.pl/~reclaw
Piotr Szuca
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, E-mail: [email protected]
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Abstract

We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

Keywords

Bolzano-Weierstrass propertyBolzano-Weierstrass theoremstatistical densitystatistical convergenceideal convergencefilter convergencesubsequenceextending idealsP-idealsP-pointsanalytic idealsmaximal ideals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 501 - 512
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Balcar, B., Hernández-Hernández, F., and Hrušák, M., Combinatorics of dense subsets of the rationals, Fundamenta Mathematical vol. 183 (2004), no. 1, pp. 59–80.CrossRefGoogle Scholar
[2]Balcerzak, Marek and Dems, Katarzyna, Some types of convergence and related Baire systems, Real Analysis Exchange, vol. 30 (2004/2005), no. 1, pp. 267–276.CrossRefGoogle Scholar
[3]Baumgartner, J. E., Taylor, A. D., and Wagon, S., Structural properties of ideals, Polska Akademia Nauk. Instytut Matematyczny. Dissertationes Mathematicae. Rozprawy Matematyczne, vol. 197 (1982), p. 95.Google Scholar
[4]Bernstein, Allen R., A new kind of compactness for topological spaces, Fundamenta Mathematicae, vol. 66 (1969/1970), pp. 185–193.CrossRefGoogle Scholar
[5]Boldjiev, B. and Malyhin, V., The sequentiality is equivalent to the F-Fréchet-Urysohn property, Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), no. 1, pp. 23–25.Google Scholar
[6]Demirci, Kamil, J-limit superior and limit inferior, Mathematical Communications, vol. 6 (2001), no. 2, pp. 165–172.Google Scholar
[7]Farah, Ilijas, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702.CrossRefGoogle Scholar
[8]Farah, Ilijas, How many Boolean algebras P(ℕ)/J are there?, Illinois Journal of Mathematics, vol. 46 (2002), no. 4, pp. 999–1033.CrossRefGoogle Scholar
[9]Fast, H., Sur la convergence statistique, Colloquium Math., vol. 2 (1951), pp. 241–244 (1952).CrossRefGoogle Scholar
[10]Fridy, J. A., Statistical limit points, Proceedings of the American Mathematical Society, vol. 118 (1993), no. 4, pp. 1187–1192.CrossRefGoogle Scholar
[11]Hernández-Hernández, F. and Hrušak, M., Cardinal invariants of analytic P-ideals.Google Scholar
[12]Just, Winfried and Krawczyk, Adam, On certain Boolean algebras P (ω)/I, Transactions of the American Mathematical Society, vol. 285 (1984), no. 1, pp. 411–429.Google Scholar
[13]Just, Winfried and Mijajlović, Žarko, Separation properties of ideals over ω, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), no. 3, pp. 267–276.CrossRefGoogle Scholar
[14]Katétov, Miroslav, Products of filters, Commentationes Mathematicae Universitatis Carolinae, vol. 9 (1968), pp. 173–189.Google Scholar
[15]Kojman, Menachem, Hindman spaces, Proceedings of the American Mathematical Society, vol. 130 (2002), no. 6, pp. 1597–1602 (electronic).CrossRefGoogle Scholar
[16]Kojman, Menachem, Van der Waerden spaces, Proceedings of the American Mathematical Society, vol. 130 (2002), no. 3, pp. 631–635 (electronic).CrossRefGoogle Scholar
[17]Kostyrko, Pavel, Šalát, Tibor, and Wilczyński, Władysław, J-convergence, Real Analysis Exchange, vol. 26 (2000/2001), no. 2, pp. 669–685.CrossRefGoogle Scholar
[18]Louveau, Alain and Veličković, Boban, A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), no. 1, pp. 255–259.CrossRefGoogle Scholar
[19]Mazur, Krzysztof, Fσ-ideals and ω1ω1*-gaps in the Boolean algebras P(ω)/I, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 138 (1991), no. 2, pp. 103–111.CrossRefGoogle Scholar
[20]Monk, J. Donald, Continuum cardinals generalized to Boolean algebras, this Journal, vol. 66 (2001), no. 4, pp. 1928–1958.Google Scholar
[21]Nuray, Fatih and Ruckle, William H., Generalized statistical convergence and convergence free spaces, Journal of Mathematical Analysis and Applications, vol. 245 (2000), no. 2, pp. 513–527.CrossRefGoogle Scholar
[22]Rudin, Walter, Homogeneity problems in the theory ofČech compactifications, Duke Mathematical Journal, vol. 23 (1956), pp. 409–419.CrossRefGoogle Scholar
[23]Šalát, Tibor, Tripathy, Binod Chandra, and Ziman, Miloš, On some properties of J-convergence, Tatra Mountains Mathematical Publications, vol. 28 (2004), pp. 279–286.Google Scholar
[24]Shelah, Saharon, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[25]Solecki, Sławomir, Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1–3, pp. 51–72.CrossRefGoogle Scholar