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.
Abstract. We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for F σδ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.
Introduction.A family of sets J ⊂ P (ω) is an ideal if it is closed under taking finite unions and subsets. Throughout this paper we assume that [ω] <ω ⊂ J, ω ∈ J, and we write Fin = [ω] <ω . In a metric space X, a sequence {x n } n<ω is J-convergent to x if (∀ε > 0) {n : (x n , x) ≥ ε} ∈ J; we write J-lim x n = x. For functions f, f n : X → Y, where Y is a metric space, define J-lim f n = f iff J-lim f n (x) = f (x) for each x ∈ X. J-lim F will denote the set of all J-limits of sequences of functions from F. For every topological space X let C(X) be the family of all continuous real-valued functions, and let B α (X) be the family of all real-valued functions on X of Baire class α < ω 1 . An ideal J is called a P-ideal if for each family {A n : n ∈ ω} ⊂ J there is A ∈ J with A n ⊂ * A (i.e. |A \ A n | < ω) for each n. Let J * = {A : ω \ A ∈ J}.It is known that J-limits can be very irregular.
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