Let R denote the set of reals and N the set of positive integers. By τ 0 we shall denote the natural topology on R. Let B(τ), K(τ), Ba(τ) denote the family of all Borel sets, meager sets and sets having the Baire property in a topologicalwhere int τ and cl τ mean the interior and closure with respect to the topology τ. If τ = τ 0 then we shall use the notation B, K and Ba, respectively. The symmetric difference of sets A, B is denoted by A B.Let Φ : τ 0 → 2 R be an operator satisfying the following conditions:Let Φ stand for the family for all operators satisfying conditions (i) − (iii).Remark 5.1. If Φ ∈ Φ then Φ(A) ⊂ cl τ 0 A for every A ∈ τ 0 .

In the paper we give and investigate the basic properties of a method for improving continuity, quasi-continuity and the Darboux property of real functions defined on a metric Baire space. This "improvement" is carried out with the use of Blumberg sets.A. Katafiasz in her doctoral dissertation [5] (Also see [6] and [7].) introduced the notion of an α-improvable discontinuous function. The main idea of this work was to examine the possibility of the "removal" of the discontinuity of real functions defined on some subsets of the line. The author suggested a certain method for removing the discontinuity and investigated the structure of functions for which this method is efficient as well as the successive steps of the procedure of removing of the discontinuity. This dissertation has inspired our team's investigations.In our paper we give another method for "improving functions". In addition we show that our method is efficient for a considerably wider class of functions than A. Katafiasz's method, and that it may also be applied, in some cases, to the improving of quasi-continuity and the Darboux property. An additional merit of our method is the fact that the "improvement" is done in one step.

The paper concerns topologies introduced in a topological space (X τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated. MSC: 54A05, 28A05Keywords

ABSTRACT. This paper presents a density type topology with respect to an extension of Lebesgue measure involving sequence of intervals tending to zero. Some properties of such topologies are investigated.Let R denote a set of real numbers, N a set of natural numbers, and λ a Lebesgue measure on R. By L we understand a family of Lebesgue measurable sets, by L a family of Lebesgue measurable null sets, and by |I| a length of an interval I. By μ we denote any complete extension of Lebesgue measure λ, by S μ a domain of function μ, and by I μ a family of μ-null sets. If A, B are families of subsets of the space X, then we use notation A B = {C ⊂ X : C = A \ B, A ∈ A, B ∈ B} and AΔB = {C ⊂ X : C = AΔB, A ∈ A, B ∈ B}, where Δ is an operation of the symmetric difference. It is well-known that if A is σ-algebra of sets in X and B is σ-ideal of sets in X, then the family AΔB is the smallest σ-algebra containing A ∪ B.It is clear that x 0 ∈ R is a density point of a set A ∈ L ifIt is equivalent toThe above condition can be written as in [8]:where {J n } n∈N is a sequence of closed intervals.

The lattices generated by the families of quasicontinuous and τ -quasicontinuous functions are described. Keywords Functions quasicontinuous · Discrete limitsWe point out that, except for the case where a topology T is specifically mentioned, all topological notions are given with respect to the natural topology.Throughout the paper, R and N will denote the sets of all real numbers and of all positive integers, respectively, I the σ -ideal of subsets of R of the first category, S the σ -field of subsets of R having the Baire property. For any a ∈ R and A ⊂ R we denote a · A = {ax : x ∈ A} and A − a = {x − a : x ∈ A}. Recall [7] that 0 is an I-density point of set A ∈ S if and only if, for every increasing sequence {n m } m∈N of positive integers, there exists a subsequence {n m p } p∈N such that χ n m p ·A∩[−1,1] (x) −→ 1 as p −→ ∞ Communicated by P. de Lucia.