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On (∆2) condition in density-type topologies

Filipczak M., Terepeta M.

Demonstratio Mathematica

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2011

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Vol. 44, nr 2

423--432

EN

We discuss properties of density-type topologies Tψ connected with condition (∆2) similar to the condition considered in the theory of Orlicz spaces. Density-type topologies Tψ introduced in [5] may not be invariant under multiplication by a number. This property is strictly connected with the condition, which we call (∆2), by analogy with well known condition introduced in Orlicz spaces. Like in the theory of Orlicz spaces, (∆2) condition causes that the considered topologies are more convenient for examination and have simpler properties. Moreover, the power functions are also of great importance as a handy instrument. Recall some basic facts. Let (Ω, Σ, μ) be a measure space and A be a family of all functions φ: [0, ∞) → [0, ∞) which are continuous, nondecreasing, such that φ(0) = 0, φ(x) > 0 for x > 0 and limx→∞ φ(x) = ∞.

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On density with respect to equivalent measures

Bartosiewicz A., Filipczak M., Poreda T.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

21-28

EN

In the paper there is disscussed a notion of a density point of a Borel subset of a metric space with respect to a Borel measure(mikro) . There are considered densities with respect to equivalent measures and density with respect to the limit of a sequence of equivalent measures.

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On the comparison of the density type topologies generated by sequences and by functions

Filipczak M., Filipczak T.

Commentationes Mathematicae

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2009

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Vol. 49, [Z] 2

161-170

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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On homeomorphisms of the density type topologies

Filipczak M., Hejduk J., Wilczyński W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2005

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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