Elżbieta Wagner-Bojakowska

On some modification of Darboux property

Abstract We introduce some family of functions f: ℝ → ℝ modifying Darboux property analogously as it was done in GRANDE, Z.: On a subclass of the family of Darboux functions, Colloq. Math. 117 (2009), 95-104, and changing approximate continuity with 𝓘-approximate continuity, i.e. continuity with respect to the 𝓘-density topology. We prove that our family is a strongly porous set in the space of Darboux functions having the Baire property and that each function from our family is quasi-continuous.

On some Modification of Świtąkowski Property

Abstract We introduce some families of functions ƒ : ℝ → ℝ modifying the Darboux property analogously as it was done by [Maliszewski, A.: On the limits of strong Świątkowski functions, Zeszyty Nauk. Politech. Łódź. Mat. 27 (1995), 87-93], replacing continuity with A-continuity, i.e., the continuity with respect to some family A of subsets in the domain. We prove that if A has (*)-property then the family DA of functions having A-Darboux property is contained and dense in the family DQ of Darboux quasi-continuous functions.

Cauchy condition for the convergence in category

It is well known that the sequence {fn}n∈N of real measurable functions converges in measure to some measurable function f if and only if {fn}n∈N is fundamental in measure. In this note we introduce the notion of sequence fundamental in category in this manner such that the sequence {fn}n∈N of real functions having the Baire property converges in category to some function f having the Baire property if and only if {fn}n∈N is fundamental in category.

Porous subsets in the space of functions having the Baire property

Abstract The comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property. We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.

Fubini Property for Microscopic Sets

Abstract In 2000, I. Recław and P. Zakrzewski introduced the notion of Fubini Property for the pair (I,J) of two σ-ideals in the following way. Let I and J be two σ-ideals on Polish spaces X and Y, respectively. The pair (I,J) has the Fubini Property (FP) if for every Borel subset B of X×Y such that all its vertical sections Bx = {y ∈ Y : (x, y) ∈ B} are in J, then the set of all y ∈ Y, for which horizontal section By = {x ∈ X : (x, y) ∈ B} does not belong to I, is a set from J, i.e., {y ∈ Y : By ∉ I} ∈ J. The Fubini property for the σ-ideal M of microscopic sets is considered and the proof that the pair (M,M) does not satisfy (FP) is given.

(B∆I,I)-saturated sets and Hamel basis

Abstract Let I be a proper σ-ideal of subsets of the real line. In a σ-field of Borel sets modulo sets from the σ-ideal I we introduce an analogue of the saturated non-measurability considered by Halperin. Properties of (B∆I,I)-saturated sets are investigated. M. Kuczma considered a problem how small or large a Hamel basis can be. We try to study this problem in the context of sets from I.

On some problem of Sierpiński and Ruziewicz concerning the superposition of measurable functions. Microscopic Hamel basis.

Abstract S. Ruziewicz and W. Sierpiński proved that each function f : ℝ → ℝ can be represented as a superposition of two measurable functions. Here, a strengthening of this theorem is given. The properties of Lusin set and microscopic Hamel bases are considered

Density topologies on the plane between ordinary and strong

Abstract Let C0 denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx→0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a measurable subset of the plane. We define the notion of a density point of A with respect to ƒ. This is a starting point to introduce the mapping Dƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping Dƒ leads to the topology Tƒ, analogously as for the density topology. The properties of the topologies Tƒ are considered.