2007
DOI: 10.2178/jsl/1185803621
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Ideal convergence of bounded sequences

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 quotie… Show more

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Cited by 55 publications
(69 citation statements)
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“…P r o o f. If I = ∅ and J = ∅ then Fin × Fin ⊂ I × J . In [7] we showed that Fin × Fin is not Fin-BW so I × J is not Fin-BW either. …”
Section: Local Versionmentioning
confidence: 84%
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“…P r o o f. If I = ∅ and J = ∅ then Fin × Fin ⊂ I × J . In [7] we showed that Fin × Fin is not Fin-BW so I × J is not Fin-BW either. …”
Section: Local Versionmentioning
confidence: 84%
“…For the discussion and applications of these properties see [7], where we examine all BW-like properties. In particular, it is known that the ideal I d of sets of density 0 is not BW, and every F σ ideal is h-Fin-BW.…”
Section: Bolzano-weierstrass Propertymentioning
confidence: 99%
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“…Ideals fulfilling the condition AP(I, Fin) are sometimes called P-ideals (see for example [2], [13] or [14]). …”
Section: Introductionmentioning
confidence: 99%