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Commentationes Mathematicae
EN In the paper we investigate density type topologies generated by functions f satisfying condition [formula] which are not generated by any sequence.
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Journal of Applied Analysis
EN A function f : R w indeksie górnym m → R satisfies the condition [wzór] (resp. [wzór]) at a point x ∈ R w indeksie górnym m if for each real ε > 0 and for each set U ∋ x belongong to Euclidean topology in R w indeksie górnym (resp. to the strong density topolgy [to the ordinary density topology]) t[...]
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Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
2005 Vol. 45, [Z] 2 151--159
EN This paper is dealing of the homeomorphisms of the density type topologies introduced in: M. Filipczak, J. Hejduk, "On topologies associated with the Lebesgue measure", Tatra Mountains Math. Publ. 28 (2004), 187-197
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Journal of Applied Analysis
2013 Vol. 19, nr 2 283--304
EN We present a further generalization of the T Ad-density topology introduced in [Real Anal. Exchange 32 (2006/07), 349–358] as a generalization of the density topology. We construct an ascending sequence [wzór] of density topologies which leads to the [wzór]-density topology including all previous to[...]
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Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
2010 Vol. 15 67--80
EN The purpose of this paper is to study the notion of a Ψ I-density point and Ψ I -density topology, generated by it analogously to the classical I-density topology on the real line. The idea arises from the note by Taylor [3] and Terepeta and Wagner-Bojakowska [2].
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Demonstratio Mathematica
EN In this article we investigate the ideal of linear sets A such that for each nonempty set U contained in the closure cl(A) of A and belonging to the density topology the intersection U D A is a nowhere dense subset of U.
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Demonstratio Mathematica
2013 Vol. 46, nr 1 197--208
EN In this paper we shall study a density-type topology generated by the convergence everywhere except for a finite set similarly as the classical density topology is generated by the convergence in measure. Among others it is shown that the set of finite density points of a measurable set need not be [...]
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Journal of Applied Analysis
EN Let (X, Tx) be a topological space and let (Y, dy) be a metric space. For a function f : X → y denote by C(f) the set of all continuity points of f and by D(f) = X\C(f) the set of all discontinuity points of f. Let C(X,Y) = {f : X → Y; f is continuous}, H(X, Y) = {f: X →Y; D{f) is countable}, H1(X,[...]
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Demonstratio Mathematica
2004 Vol. 37, nr 2 439--445
EN This paper contains some results about density with respect to a sequence and an extension of the Lebesgue measure. There are some properties of topologies associated with such density point.
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Demonstratio Mathematica
EN In this article we investigate the products of two unilaterally approximately continuous and simultaneously approximately regulated functions. In particular we prove some necessary conditions satisfied by the products of two such functions and a sufficient condition ensuring that a function is the p[...]
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Journal of Applied Analysis
2002 Vol. 8, nr 2 201-219
EN The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied.
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Demonstratio Mathematica
2004 Vol. 37, nr 3 679--688
EN A point x C X is called universal element for a family phi of functions from X to y if the set {f(x)\f 6 phi} is dense in Y. In this article we show that every residual G- set in a completely regular space X (every residual set in R ) is the set of all universal elements for some family of continuou[...]
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Demonstratio Mathematica
2005 Vol. 38, nr 3 533--550
EN Some special notions of approximate quasicontinuity on Rm and the uniform, pointwise, transfinite and the discrete convergence of sequences of such functions are investigated.
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