On a strong generalized topology with respect to the outer Lebesgue measure

Abstract

The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that this family is a strong generalized topology. In the paper some properties of the strong generalized topology connected with density points with respect to the outer Lebesgue measure will be presented. Moreover, among others, some characterizations of the families of meager sets and compact sets in this space will be given.

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Acknowledgements

The authors are very grateful to the referee for a number of helpful suggestions for improvement of the paper. Particularly, for valuable information concerning the beginning of research on generalized topologies.

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Correspondence to A. Loranty.

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Hejduk, J., Loranty, A. On a strong generalized topology with respect to the outer Lebesgue measure. Acta Math. Hungar. 163, 18–28 (2021). https://doi.org/10.1007/s10474-020-01124-4

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Key words and phrases

  • generalized topology
  • density topology
  • (outer) Lebesgue measure
  • Bernstein set

Mathematics Subject Classification

  • 54A05
  • 28A12