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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: rfilipow@math.univ.gda.pl, 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: nmrozek@math.univ.gda.pl
Ireneusz Recław
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, E-mail: reclaw@math.univ.gda.pl, 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: pszuca@radix.com.pl
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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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