Ideal convergence of bounded sequences

Publisher: Cambridge University Press

E-ISSN: 1943-5886|72|2|501-512

ISSN: 0022-4812

Source: The Journal of Symbolic Logic, Vol.72, Iss.2, 2007-06, pp. : 501-512

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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.