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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 28, 2014

Pointwise density topology

Magdalena Górajska
From the journal Open Mathematics

Abstract

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.

References

[1] Goffman C., Neugebauer C. J., Nishura T., The density topology and approximate continuity, Duke Math. J., 1961, 28, 497-506.10.1215/S0012-7094-61-02847-2Search in Google Scholar

[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 .Search in Google Scholar

[3] Hejduk J., On density topologies with respect to invariant σ-ideals, J. Appl. Anal., 2002, 8(2), 201-219.10.1515/JAA.2002.201Search in Google Scholar

[4] 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

[5] Nowik A., Vitali set can be homeomorphic to its complement, Acta Math. Hungar., 2007, 115(1-2), 145-154.10.1007/s10474-007-5222-7Search in Google Scholar

[6] Oxtoby J. C., Measure and category, Springer Verlag, New York-Heidelberg-Berlin, 1980.Search in Google Scholar

[7] 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

[8] Wilczy´ nski W., Aversa V., Simple density topology, Rend. Circ. Mat. Palermo (2), 2004, 53, 344-352.10.1007/BF02875727Search in Google Scholar

[9] Wilczy´ nski W., Density topologies, Scientific Bulletyn Of Chełm Section of Mathematics And Computer Science, 2007, 1, 223-227. Search in Google Scholar

Received: 2013-1-2
Accepted: 2014-7-8
Published Online: 2014-10-28
Published in Print: 2015-1-1

© 2015 Magdalena Górajska

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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