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We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class $$\pmb \Pi ^0_3$$ Π 3 0 which is analogous to the measure case. We also introduce the notion of the... more
We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class $$\pmb \Pi ^0_3$$ Π 3 0 which is analogous to the measure case. We also introduce the notion of the Baire category density point of a subset with the Baire property of the Cantor space, and we prove that it generates a lower density operator with Borel values of class $$\pmb \Pi ^0_3$$ Π 3 0 .
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In this paper we will consider 𝓙-density topology connected with a sequence 𝓙 of closed intervals tending to 0 and a 𝓙-approximately continuous function associated with that kind of density points. It will be the continuation of the... more
In this paper we will consider 𝓙-density topology connected with a sequence 𝓙 of closed intervals tending to 0 and a 𝓙-approximately continuous function associated with that kind of density points. It will be the continuation of the investigations started in “𝓙-approximately continuous functions” by J. Hejduk, A. Loranty and R. Wiertelak published in Tatra Mountains Mathematical Publications. In particular, we will show that topology generated by the sequence 𝓙 from the special family 𝔍 α is always completely regular.
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This note i s concerned with 6 i d e a l s in a measurable space (X, S) , such that convergence with respect to a (5-ideal 3 induces the Frechet topology on the space of a l l 5 v 3 -measurable r e a l funct ions , where S ' 3 i s the... more
This note i s concerned with 6 i d e a l s in a measurable space (X, S) , such that convergence with respect to a (5-ideal 3 induces the Frechet topology on the space of a l l 5 v 3 -measurable r e a l funct ions , where S ' 3 i s the smallest d f i e l d conta ining both <5 and 3 . These ¿ i d e a l s are ca l led ¿ t o p o l o g i c a l . Suppose that we are given a nonempty set X. In a l l that follows below, we consider d f i e l d s and didea ls of subsets of X. To begin with, l e t ua r e c a l l two d e f i n i t i o n s . D e f i n i t i o n 1. ( c f . [ 4 ] ) . l e t 3 be a ¿ i d e a l . We say tnat a property holds 3-almost everywhere on X (abbr. 3 a . e . ) i f the set of a l l points which do not have t h i s property belongs to 3 . D e f i n i t i o n 2. ( c f . [ 4 ] ) . We say that a s e quence { f n } n e N of r e a l funct ions defined on X converges with respect to a 6 i d e a l 3 to a funct ion f ( a b b r « { f n } i T ^ £ i f
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This paper contains the properties of continuous functions equipped with the$\mathcal{J}$-density topology or natural topology in the domain and the range.
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This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a... more
This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.
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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.
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This paper deals with the abstract density topologies in the family of Lebesgue measurable sets generated by an operator similar to the density lower operator defined in the family of measurable sets.
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The paper concerns some local properties of the sets with pointwise density points in terms of measure and category on the real line. We also construct nonmeasurable and not having the Baire property sets with pointwise density point.
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This paper presents the properties of continuous functions equipped with the J-density topology or natural topology in the domain and the range.
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In this paper, density-like points and density-like topology connected with a sequence I = {In}n∊ℕ of closed intervals tending to 0 will be considered. We introduce the notion of an I -approximately continuous function associated with... more
In this paper, density-like points and density-like topology connected with a sequence I = {In}n∊ℕ of closed intervals tending to 0 will be considered. We introduce the notion of an I -approximately continuous function associated with this kind of density points. Moreover, we present some properties of these functions and we demonstrate their connection with continuous functions with respect to this kind of density topology.
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The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously,... more
The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.
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The paper presents a pointwise density topology with respect to admissible σ-algebras on the real line. The properties of such topologies, including the separation axioms, are studied
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The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager... more
The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager sets. The regularity property of such topologies is investigated.
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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... more
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.
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The first part of the paper contains some ideas of the density topologies in the measurable spaces. The second part is devoted to the difference between measure and category for the abstract density space related to the separation axioms