Cardinal Invariants of AnalyticP-Ideals

Abstract

Abstract. We study the cardinal invariants of analytic P-ideals, concentrating on the ideal Z of asymptotic density zero. Among other results we prove min{b, cov (N)} ≤ cov * (Z) ≤ max{b, non(N)}.

Cardinal coefficients associated to certain orders on ideals

“…• the minimal cardinality of a family of elements of the dual filter I * which does not have a pseudo-intersection in I * (p(I * ), see [5]; add * (I), see [8]); • the minimal cardinality of a family of elements of the dual filter I * which does not have a pseudo-intersection (χ p(I * ), see [5]; cov * (I), see [8]) ; • the minimal cardinality of a family A with the I-strong finite intersection property (every finite subfamily has an intersection outside I) without a set outside I which is almost included (in the sense of I) in every member of A (p I , see [9]; p(I), see [4]). …”

Adding ultrafilters by definable quotients

“…The following proposition, which can be found in [6], shows that we can replace tall summable ideal with tall analytic P ideal in the above theorem.…”

“…(iii) (Katětov ordering, [11]) I ≤ K J if there is a function f : ω → ω such that preimages of I-small sets are J -small. (iv) (Katětov-Blass ordering, [6]) I ≤ KB J if I ≤ K J and the witnessing function can be chosen to be finite-to-one.…”

On sequential properties of Banach spaces, spaces of measures and densities

“…One can check that k * b and that k * < b is relatively consistent, basing on some results on the cardinal number b, see Blass [3]. Ideals of the form {A ⊆ ω : d ϕ (A) = 0}, where ϕ : ω → R + , are sometimes called the Erdős-Ulam ideals; cardinal invariants of such ideals on ω are considered by Hrusak [11] and Farkas & Soukup [8].…”