### 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:
Primary Subjects: 40A05
Secondary Subjects: 26A03, 54A20
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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

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