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.
@article{bwmeta1.element.doi-10_1515_math-2015-0008,
author = {Magdalena G\'orajska},
title = {Pointwise density topology},
journal = {Open Mathematics},
volume = {13},
year = {2015},
zbl = {1310.28002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0008}
}
Magdalena Górajska. Pointwise density topology. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0008/
[1] Goffman C., Neugebauer C. J., Nishura T., The density topology and approximate continuity, Duke Math. J., 1961, 28, 497-506.
[2] Górajska M., Wilczy´ nski W., Density topology generated by the convergence everywhere except on a finite set, Demonstratio Math. 2013, 45(1),197-208 . | Zbl 1280.54005
[3] Hejduk J., On density topologies with respect to invariant σ-ideals, J. Appl. Anal., 2002, 8(2), 201-219. | Zbl 1029.28001
[4] Kuczma M., An introduction to the theory of functional equation and inequalities, 2en ed., Birkhäuser Verlag AG, Basel-Boston- Berlin, 2009.
[5] Nowik A., Vitali set can be homeomorphic to its complement, Acta Math. Hungar., 2007, 115(1-2), 145-154.[WoS] | Zbl 1136.03031
[6] Oxtoby J. C., Measure and category, Springer Verlag, New York-Heidelberg-Berlin, 1980. | Zbl 0435.28011
[7] Poreda W., Wagner-Bojakowska E., Wilczy´ nski W., A category analogue of the density topology, Fund.Math. 1985, 125, 167-173. | Zbl 0613.26002
[8] Wilczy´ nski W., Aversa V., Simple density topology, Rend. Circ. Mat. Palermo (2), 2004, 53, 344-352. | Zbl 1194.26002
[9] Wilczy´ nski W., Density topologies, Scientific Bulletyn Of Chełm Section of Mathematics And Computer Science, 2007, 1, 223-227.