Ireneusz Recław

Spaces not distinguishing pointwise and quasinormal convergence of real functions

Abstract A topological space X is said to be a wQN-space if from every sequence of continuous real functions converging pointwise to zero on X one can choose a quasinormally converging sub- sequence. Some properties of this and related notions are studied.

Restrictions to continuous functions and Boolean algebras

We show that every Borel function f: R → R is continuous on a set A ¬∈ J if B(R)/J is weakly distributive. We also show that CCC is not sufficient. We investigate some other conditions considering the problem of restrictions to continuous functions

Ranks of F-limits of filter sequences

Abstract We give an exact value of the rank of an F -Fubini sum of filters for the case where F is a Borel filter of rank 1 . We also consider F -limits of filters F i , which are of the form lim F F i = { A ⊂ X : { i ∈ I : A ∈ F i } ∈ F } . We estimate the ranks of such filters; in particular, we prove that they can fall to 1 for F as well as for F i of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.

On the Difference Property of Borel Measurable and \(s\)-Measurable Functions

We prove that the class of (s)-measurable functions does not have the difference property. We show also under CH that there is a function with Borel differences but of unlimited Baire class. It solves a problem of M. Laczkovich.

On cardinal invariants for CCC -ideals

We show several results about cardinal invariants for σ-ideals of the reals. In particular we show that for every CCC σ-ideal on the real line p ≤ cof(J). In this paper we will consider σ-ideals on the real line with Borel basis; i.e., every element from the σ-ideal is contained in a Borel set from the σ-ideal. We assume also that every singleton is in the σ-ideal. We say that σ-ideal J is CCC (countable chain condition) if the quotient Boolean algebra B(R)/J is CCC, where B(R) is the σ-algebra of all Borel sets of the reals. We say that a σ-ideal J is invariant if for each A ∈ J and t ∈ R , A + t ∈ J and −A ∈ J . We define the following cardinal invariants of J : cov(J ) = min{|F | : F ⊂ J and ⋃F = R}, add(J ) = min{|F | : F ⊂ J and ⋃F 6∈ J }, non(J ) = min{|X | : X ⊂ R and X 6∈ J }, cof(J ) = min{|F | : F ⊂ J and ∀X∈J ∃Y ∈FX ⊂ Y }, covl(J ) = min{|F | : F ⊂ J and ∃B 6∈JBorelB ⊂ ⋃ F}. Observe that covl(J ) can be smaller than cov(J ), for example, in some models of set theory for the σ-ideal generated by meagre sets on (−∞, 0] and null sets on [0,∞). p = min{|F | : F ⊂ [ω] such that for every finite subset F0 of F , | ⋂ F0| = ω and there is no A ∈ [ω] such that for every B ∈ F, |B \A| < ω} (see [D]). Definition. X ⊂ R is a Q-set if every subset of X is relatively Gδ in X. Theorem 1. Let J ⊂ P (R) be a σ-ideal with Borel basis and CCC in B(R). Then p ≤ cof(J ). Proof. In fact, in the proof we use only that every set of size less than p is a Qset. Assume that cof(J ) < p. We know that every set of size cof(J ) is a Q-set (see [MS]). Let X ⊂ R with X 6∈ J of size cof(J ). Let B be a Borel set such that X ⊂ B and for each Borel set D ⊂ (B \ X) , D ∈ J . Such a set exists by CCC. Observe that B \ X 6∈ J . Otherwise, there is a Borel set B1 ⊃ (B \ X) with B1 ∈ J ; thus B \ B1 ⊂ X and B \ B1 is Borel and B \ B1 6∈ J which is Received by the editors March 3, 1995 and, in revised form, September 16, 1996. 1991 Mathematics Subject Classification. Primary 04A20; Secondary 03E35.