In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized t... more In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.
The paper deals with the strong porosity of some families of real functions continuous with respe... more The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.
The classical density topology is the topology generated by the lower density operator connected ... more The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that this family is a strong generalized topology. In the paper some properties of the strong generalized topology connected with density points with respect to the outer Lebesgue measure will be presented. Moreover, among others, some characterizations of the families of meager sets and compact sets in this space will be given.
Tatra mountains mathematical publications, Aug 10, 2015
In this paper density-like points and density-like topology connected with a sequence $\mathcal{J... more In this paper density-like points and density-like topology connected with a sequence $\mathcal{J} = \{J}_n n \in \mathbb{N}$ of closed intervals tending to $0$ will be considered. We will introduce the notion of an $\mathcal{J}$-approximately continuous function associated with that kind of density points. Moreover, we will present some properties of these functions and show their connection with continuous functions with respect to such density topology.
The aim of this paper is the investigation of topologies generated by operators related to a comp... more The aim of this paper is the investigation of topologies generated by operators related to a complete extension of the Lebesgue measure over the real line. Some properties of such topologies provide their structure and separation axioms.
Through the paper we shall use the standard notation: R will be the set of real numbers, L the fa... more Through the paper we shall use the standard notation: R will be the set of real numbers, L the family of Lebesgue measurable subsets of R and l(A) the Lebesgue measure of a measurable set A. By N we shall denote the set of all positive integers ans by S the family of all ...
In the paper, some properties of functions continuous with respect to a density type strong gener... more In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.
Hacettepe Journal of Mathematics and Statistics, 2020
In the paper we concentrate on a generalized topological space generated by a density type operat... more In the paper we concentrate on a generalized topological space generated by a density type operator on a measurable space. The properties of such generalized topological space are investigated. Moreover, the properties of nowhere dense sets, meager sets and compact sets in this generalized topological space are studied.
In the paper we concentrate on lower, almost-lower and semi-lower density operators on measurable... more In the paper we concentrate on lower, almost-lower and semi-lower density operators on measurable spaces. The existence of maximal element in the families of such operators is investigated. Moreover, we consider topologies generated by the above operators. Among others the existence of the greatest of such topologies (with respect to the inclusion) is studied.
Abstract The paper presents a new family of density-type topologies. The pointwise density topolo... more Abstract The paper presents a new family of density-type topologies. The pointwise density topology introduced in [4] is a motivation to consider topologies with respect to a fixed sequence. We study several properties of such topologies, in particular we discuss the separation axioms. Moreover, we apply our results to show that the pointwise density topology on the real line is not a regular space.
Some kind of abstract density topology in a topological Baire space is considered. The semiregula... more Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.
In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized t... more In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.
The paper deals with the strong porosity of some families of real functions continuous with respe... more The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.
The classical density topology is the topology generated by the lower density operator connected ... more The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that this family is a strong generalized topology. In the paper some properties of the strong generalized topology connected with density points with respect to the outer Lebesgue measure will be presented. Moreover, among others, some characterizations of the families of meager sets and compact sets in this space will be given.
Tatra mountains mathematical publications, Aug 10, 2015
In this paper density-like points and density-like topology connected with a sequence $\mathcal{J... more In this paper density-like points and density-like topology connected with a sequence $\mathcal{J} = \{J}_n n \in \mathbb{N}$ of closed intervals tending to $0$ will be considered. We will introduce the notion of an $\mathcal{J}$-approximately continuous function associated with that kind of density points. Moreover, we will present some properties of these functions and show their connection with continuous functions with respect to such density topology.
The aim of this paper is the investigation of topologies generated by operators related to a comp... more The aim of this paper is the investigation of topologies generated by operators related to a complete extension of the Lebesgue measure over the real line. Some properties of such topologies provide their structure and separation axioms.
Through the paper we shall use the standard notation: R will be the set of real numbers, L the fa... more Through the paper we shall use the standard notation: R will be the set of real numbers, L the family of Lebesgue measurable subsets of R and l(A) the Lebesgue measure of a measurable set A. By N we shall denote the set of all positive integers ans by S the family of all ...
In the paper, some properties of functions continuous with respect to a density type strong gener... more In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.
Hacettepe Journal of Mathematics and Statistics, 2020
In the paper we concentrate on a generalized topological space generated by a density type operat... more In the paper we concentrate on a generalized topological space generated by a density type operator on a measurable space. The properties of such generalized topological space are investigated. Moreover, the properties of nowhere dense sets, meager sets and compact sets in this generalized topological space are studied.
In the paper we concentrate on lower, almost-lower and semi-lower density operators on measurable... more In the paper we concentrate on lower, almost-lower and semi-lower density operators on measurable spaces. The existence of maximal element in the families of such operators is investigated. Moreover, we consider topologies generated by the above operators. Among others the existence of the greatest of such topologies (with respect to the inclusion) is studied.
Abstract The paper presents a new family of density-type topologies. The pointwise density topolo... more Abstract The paper presents a new family of density-type topologies. The pointwise density topology introduced in [4] is a motivation to consider topologies with respect to a fixed sequence. We study several properties of such topologies, in particular we discuss the separation axioms. Moreover, we apply our results to show that the pointwise density topology on the real line is not a regular space.
Some kind of abstract density topology in a topological Baire space is considered. The semiregula... more Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.
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