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.
 Górajska M., Wilczy´ nski W., Density topology generated by the convergence everywhere except on a finite set, Demonstratio Math. 2013, 45(1),197-208 .Search in Google Scholar
 Kuczma M., An introduction to the theory of functional equation and inequalities, 2en ed., Birkhäuser Verlag AG, Basel-Boston- Berlin, 2009.Search in Google Scholar
 Oxtoby J. C., Measure and category, Springer Verlag, New York-Heidelberg-Berlin, 1980.Search in Google Scholar
 Poreda W., Wagner-Bojakowska E., Wilczy´ nski W., A category analogue of the density topology, Fund.Math. 1985, 125, 167-173.Search in Google Scholar
 Wilczy´ nski W., Density topologies, Scientific Bulletyn Of Chełm Section of Mathematics And Computer Science, 2007, 1, 223-227. Search in Google Scholar
© 2015 Magdalena Górajska
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