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A category Ψ-density topology

100%
Wilczyński W., Wojdowski W.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1057-1066
EN
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 regular open representation of A.
2
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On \(\mathbb{I}\)-approximate limits and \(\mathbb{I}\)-approximate smoothness

80%
Zduńczyk R.
Commentationes Mathematicae
|
2006
|
tom 46
|
nr 1
EN
In this paper we present some results based on slightly modified idea of the \(\mathbb{I}\)-density introduced by Władysław Wilczyński. Some theorems are generalized versions of results from [2] and [3]. We investigate properties of functions from \(\mathbb{R}^X\), where \(X\) is supplied with the \(\mathbb{I}\)-density. We try to free our considerations from the assumption of Baire property, or measurability. In some cases this is not done yet. Star-marked statements still need that assumption, proofs presented here are done for Baire property, but it is possible to adapt them to measure. \(\mathbb{I}\)-density itself does not require any structure of considered space but a metric vector space over \(\mathbb{R}\). However, in last section we confine ourselves to \(\mathbb{R}\), for we make use of \(\mathbb{R}\)’s structure for simplicity. To find more about related topics see [4], [5], more bibliography one can find in [1] and [5].
3
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Pointwise density topology

80%
Górajska M.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
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 density points of any Borel set is Lebesgue measurable.
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