Journal of Symbolic Logic

Ideal convergence of bounded sequences

Rafał Filipów, Recław Ireneusz, Mrożek Nikodem, and Szuca Piotr

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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.

Article information

J. Symbolic Logic Volume 72, Issue 2 (2007), 501-512.

First available in Project Euclid: 30 July 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 40A05: Convergence and divergence of series and sequences
Secondary: 26A03: Foundations: limits and generalizations, elementary topology of the line 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

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


Filipów, Rafał; Nikodem, Mrożek; Ireneusz, Recław; Piotr, Szuca. Ideal convergence of bounded sequences. J. Symbolic Logic 72 (2007), no. 2, 501--512. doi:10.2178/jsl/1185803621.

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