Cardinal coefficients associated to certain orders on ideals

Abstract

We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak * topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain ordering on ideals.

Ideal approach to convergence in functional spaces

“…By J. Gerlits and Zs. Nagy [17] and P. Simon and B. Tsaban [44], we obtain that all the notions in Diagram 1 have the same uniformity number, namely the pseudointersection number p. Consequently, for the pseudointersection number p K (J ) introduced by P. Borodulin-Nadzieja and B. Farkas [5], we have the following result, see Corollary 10.1.…”

“…Let β < α. By part (1), we have Fin β M ≤ K Fin α M , and by part (5), it is enough to show that Fin α ≤ K Fin β . However, this follows by results by M. Katětov [26]: 9 He has shown in his Theorem 7.1 that considering all pointwise limits of all continuous functions on a topological space with respect to Fin ξ , one obtains exactly ξ-th Baire class of functions.…”

“…Thus we distinguish an [Ω, I-Γ]-space for many of the pairs of ideals among standard critical ideals considered in [6,7,19,20], see Corollaries 6.5, 8.4, and Diagram 3. Moreover, we study similar variation of Fréchet-Urysohn property of C p (X), denoted [Ω 0 , I-Γ 0 ]-space, and introduced by P. Borodulin-Nadzieja and B. Farkas [5]. We distinguish also these properties (Theorem 9.5).…”

“…According to [5,47], C p (X) possesses J -Fréchet-Urysohn property, shortly [Ω 0 , J -Γ 0 ], 20 if for every A ⊆ C p (X) and f ∈ A there is a sequence in A which J -converges to f , 21 which means that there exists a sequence a n : n ∈ ω of elements of A such that {n ∈ ω : a n ∈ O} ∈ J for any open neighbourhood O of f . We summarize the relations between properties [Ω 0 , J -Γ 0 ] for various standard critical ideals J in Diagram 4 which is similar to Diagram 3.…”

Homogeneous ideals on countable sets

“…Let us recall that if J 1 is a dense ideal, then J 1 J 2 if and only if there is a 1 − 1 function f : [2] or [3]). Therefore, by the above fact we get that I|A I|B.…”