Ideal convergence of bounded sequences
Rafał Filipów, Recław Ireneusz, Mrożek Nikodem, and Szuca Piotr
Source: J. Symbolic Logic Volume 72, Issue 2
(2007), 501-512.
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.
First Page:
Show
Hide
Keywords: Bolzano-Weierstrass property; Bolzano-Weierstrass theorem; statistical density; statistical convergence; ideal convergence; filter convergence; subsequence; extending ideals; P-ideals; P-points; analytic ideals; maximal ideals
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1185803621
Digital Object Identifier: doi:10.2178/jsl/1185803621
Mathematical Reviews number (MathSciNet): MR2320288
Zentralblatt MATH identifier: 1123.40002
References
B. Balcar, F. Hernández-Hernández, and M. Hrušák, Combinatorics of dense subsets of the rationals, Fundamenta Mathematicae, vol. 183 (2004), no. 1, pp. 59--80.
Mathematical Reviews (MathSciNet): MR2098150
Marek Balcerzak and Katarzyna Dems, Some types of convergence and related Baire systems, Real Analysis Exchange, vol. 30 (2004/05), no. 1, pp. 267--276.
Mathematical Reviews (MathSciNet): MR2127531
Project Euclid: euclid.rae/1122482132
J. E. Baumgartner, A. D. Taylor, and S. Wagon, Structural properties of ideals, Polska Akademia Nauk. Instytut Matematyczny. Dissertationes Mathematicae. Rozprawy Matematyczne, vol. 197 (1982), p. 95.
Mathematical Reviews (MathSciNet): MR687276
Allen R. Bernstein, A new kind of compactness for topological spaces, Fundamenta Mathematicae, vol. 66 (1969/1970), pp. 185--193.
Mathematical Reviews (MathSciNet): MR251697
B. Boldjiev and V. Malyhin, The sequentiality is equivalent to the $\mathcal F$-Fréchet-Urysohn property, Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), no. 1, pp. 23--25.
Mathematical Reviews (MathSciNet): MR1056166
Kamil Demirci, $\mathcalI$-limit superior and limit inferior, Mathematical Communications, vol. 6 (2001), no. 2, pp. 165--172.
Mathematical Reviews (MathSciNet): MR1908336
Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702.
Mathematical Reviews (MathSciNet): MR1711328
--------, How many Boolean algebras $\mathcalP(\mathbbN)/\mathcalI$ are there?, Illinois Journal of Mathematics, vol. 46 (2002), no. 4, pp. 999--1033.
Mathematical Reviews (MathSciNet): MR1988247
H. Fast, Sur la convergence statistique, Colloquium Math., vol. 2 (1951), pp. 241--244 (1952).
Mathematical Reviews (MathSciNet): MR48548
Zentralblatt MATH: 0044.33605
J. A. Fridy, Statistical limit points, Proceedings of the American Mathematical Society, vol. 118 (1993), no. 4, pp. 1187--1192.
Mathematical Reviews (MathSciNet): MR1181163
Digital Object Identifier: doi:10.2307/2160076
Zentralblatt MATH: 0776.40001
F. Hernández-Hernández and M. Hrušak, Cardinal invariants of analytic P-ideals.
Winfried Just and Adam Krawczyk, On certain Boolean algebras $\mathcalP(\omega )/I$, Transactions of the American Mathematical Society, vol. 285 (1984), no. 1, pp. 411--429.
Mathematical Reviews (MathSciNet): MR748847
Digital Object Identifier: doi:10.2307/1999489
JSTOR: links.jstor.org
Zentralblatt MATH: 0519.06011
Winfried Just and Žarko Mijajlović, Separation properties of ideals over $\omega$, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), no. 3, pp. 267--276.
Mathematical Reviews (MathSciNet): MR894026
Miroslav Katětov, Products of filters, Commentationes Mathematicae Universitatis Carolinae, vol. 9 (1968), pp. 173--189.
Mathematical Reviews (MathSciNet): MR250257
Menachem Kojman, Hindman spaces, Proceedings of the American Mathematical Society, vol. 130 (2002), no. 6, pp. 1597--1602 (electronic).
Mathematical Reviews (MathSciNet): MR1887003
Digital Object Identifier: doi:10.1090/S0002-9939-01-06238-4
JSTOR: links.jstor.org
--------, Van der Waerden spaces, Proceedings of the American Mathematical Society, vol. 130 (2002), no. 3, pp. 631--635 (electronic).
Mathematical Reviews (MathSciNet): MR1866012
Digital Object Identifier: doi:10.1090/S0002-9939-01-06116-0
JSTOR: links.jstor.org
Pavel Kostyrko, Tibor Šalát, and Władysław Wilczyński, $\mathcalI$-convergence, Real Analysis Exchange, vol. 26 (2000/01), no. 2, pp. 669--685.
Alain Louveau and Boban Veličković, A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), no. 1, pp. 255--259.
Mathematical Reviews (MathSciNet): MR1169042
Digital Object Identifier: doi:10.2307/2160193
JSTOR: links.jstor.org
Zentralblatt MATH: 0794.04002
Krzysztof Mazur, $F_\sigma$-ideals and $\omega_1\omega_1^*$-gaps in the Boolean algebras $P(\omega)/I$, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 138 (1991), no. 2, pp. 103--111.
Mathematical Reviews (MathSciNet): MR1124539
J. Donald Monk, Continuum cardinals generalized to Boolean algebras, Journal of Symbolic Logic, vol. 66 (2001), no. 4, pp. 1928--1958.
Mathematical Reviews (MathSciNet): MR1877033
Digital Object Identifier: doi:10.2307/2694986
Project Euclid: euclid.jsl/1183746636
Fatih Nuray and William H. Ruckle, Generalized statistical convergence and convergence free spaces, Journal of Mathematical Analysis and Applications, vol. 245 (2000), no. 2, pp. 513--527.
Mathematical Reviews (MathSciNet): MR1758553
Digital Object Identifier: doi:10.1006/jmaa.2000.6778
Zentralblatt MATH: 0955.40001
Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Mathematical Journal, vol. 23 (1956), pp. 409--419.
Mathematical Reviews (MathSciNet): MR80902
Digital Object Identifier: doi:10.1215/S0012-7094-56-02337-7
Project Euclid: euclid.dmj/1077466953
Zentralblatt MATH: 0073.39602
Tibor Šalát, Binod Chandra Tripathy, and Miloš Ziman, On some properties of $\mathcalI$-convergence, Tatra Mountains Mathematical Publications, vol. 28 (2004), pp. 279--286.
Mathematical Reviews (MathSciNet): MR2087000
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.
Mathematical Reviews (MathSciNet): MR675955
Zentralblatt MATH: 0495.03035
Sławomir Solecki, Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1-3, pp. 51--72.
Mathematical Reviews (MathSciNet): MR1708146
Zentralblatt MATH: 0932.03060
Digital Object Identifier: doi:10.1016/S0168-0072(98)00051-7
