Displaying similar documents to “Category theorems concerning Z-density continuous functions”

A category Ψ-density topology

Władysław Wilczyński, Wojciech Wojdowski (2011)

Open Mathematics

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Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the...

On topologies generated by some operators

Katarzyna Flak, Jacek Hejduk (2013)

Open Mathematics

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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 respect to such topologies are investigated.

Pointwise density topology

Magdalena Górajska (2015)

Open Mathematics

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The paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise...

Analytic functions are ℐ -density continuous

Krzysztof Ciesielski, Lee Larson (1994)

Commentationes Mathematicae Universitatis Carolinae

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A real function is ℐ -density continuous if it is continuous with the ℐ -density topology on both the domain and the range. If f is analytic, then f is ℐ -density continuous. There exists a function which is both C ∞ and convex which is not ℐ -density continuous.