Descriptive set-theoretical properties of an abstract density operator

Szymon Gła̧b

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 732-740
  • ISSN: 2391-5455

Abstract

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Let 𝒦 (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that { K ∈ 𝒦 ( ℝ ) : ∀ x ∈ K ( d + ( x , K ) = 1 o r d - ( x , K ) = 1 ) } is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

How to cite

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Szymon Gła̧b. "Descriptive set-theoretical properties of an abstract density operator." Open Mathematics 7.4 (2009): 732-740. <http://eudml.org/doc/269147>.

@article{SzymonGła2009,
abstract = {Let \[ \mathcal \{K\} \] (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that \[ \lbrace K \in \mathcal \{K\}(\mathbb \{R\}):\forall \_x \in K(d^ + (x,K) = 1ord^ - (x,K) = 1)\rbrace \] is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.},
author = {Szymon Gła̧b},
journal = {Open Mathematics},
keywords = {Density; Porosity; Coanalytic set; ⊓11-complete set; density; porosity; coanalytic set; -complete set},
language = {eng},
number = {4},
pages = {732-740},
title = {Descriptive set-theoretical properties of an abstract density operator},
url = {http://eudml.org/doc/269147},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Szymon Gła̧b
TI - Descriptive set-theoretical properties of an abstract density operator
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 732
EP - 740
AB - Let \[ \mathcal {K} \] (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that \[ \lbrace K \in \mathcal {K}(\mathbb {R}):\forall _x \in K(d^ + (x,K) = 1ord^ - (x,K) = 1)\rbrace \] is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.
LA - eng
KW - Density; Porosity; Coanalytic set; ⊓11-complete set; density; porosity; coanalytic set; -complete set
UR - http://eudml.org/doc/269147
ER -

References

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  11. [11] Wilczyński W., A generalization of density topology, Real Anal. Exchange, 1982/83, 8, 16–20 
  12. [12] Zanyček L., Porosity and σ-porosity, Real Anal. Exchange, 1987/88, 13(2), 314–350 
  13. [13] Zanyček L., On σ-porous sets in abstract spaces, Abstr. Appl. Anal., 2005, 5, 509–534 
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  15. [15] Zelený M., Descriptive properties of σ-porous sets, Real Anal. Exchange, 2004/05, 30(2), 657–674 

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