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 quotient Boolean algebra has a countably splitting family.

Ideal version of Ramsey’s theorem

“…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. …”

“…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.…”

“…Moreover, there is no connection between Fin-BW and locally selective ideals. Indeed, NWD(Q) is not Fin-BW (see [7]) but it is locally selective. On the other hand, I 1/n is Fin-BW and is not locally selective.…”

“…Here, as in the case of the local version, the second implication of the above proposition cannot be reversed. The ideal I 1/n is h-Fin-BW (see [7]) but is not h-Mon. Whereas the second implication is in fact an equivalence as is shown in Theorem 3.16.…”

IK-Cauchy functions

“…Ideals fulfilling the condition AP(I, Fin) are sometimes called P-ideals (see for example [2], [13] or [14]). …”