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**Corrigendum in: a generalization of density topology and on generalization of the density topology on the real line.**
*(English)*
Zbl 1397.54006

Let \(\mathcal A_d\) be a family of measurable subsets of \([-1,1]\) that have density one at the point \(0\) (here \(d\) means the Lebesgue real measure). In this short note, the author provides a new definition of \(\mathcal A_d\)-density point of a measurable set \(A\subset \mathbb R\), inspired by the notion of segment density point introduced in his paper [the author, in: Real functions, density topology and related topics. Dedicated to Professor Władysław Wilczyński on the occasion of his 65th birthday. Łódź: Wydawnictwo Uniwersytetu Łódzkiego. 29–36 (2011; Zbl 1236.54004)]. With this new definition, all the results presented in his papers [Real Anal. Exch. 32, 349–358 (2007; Zbl 1135.28001)] and [Real Anal. Exch. 33, 199–214 (2008; Zbl 1151.54005)] stay valid and, even, some proofs of the mentioned papers can be shorter. It is worth pointing out that this new definition allows the author to fill a gap in the above mentioned [2007, loc. cit.], where it was erroneously established that a certain associate operator \(\Phi_{\mathcal A_d}\) is monotonic. In the present paper, the author gives a counterexample to this monotonic property for the ancient definition by finding two measurable sets \(A\subset E\) for which \(0\in\Phi_{\mathcal A_d}(A)\setminus \Phi_{\mathcal A_d}(E).\) Nevertheless, with the new definition now the operator is already a lower density operator.

Reviewer: Antonio Linero Bas (Murcia)

### MSC:

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

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\textit{W. Wojdowski}, Real Anal. Exch. 37, No. 2, 499--502 (2012; Zbl 1397.54006)

### References:

[1] | W. Wojdowski, A generalization of density topology , Real Anal. Exchange,Vol. 32(2) , (2006/2007), 1-10. |

[2] | W. Wojdowski, On a generalization of the density topology on the real line , Real Anal. Exchange,Vol. 33(1) , (2007/2008), 201-216. · Zbl 1151.54005 |

[3] | W. Wojdowski, A topology stronger than the Lebesgue density topology , in: Real functions, density topology and related topics (M. Filipczak, E. Wagner-Bojakowska eds), Łódź University Press. 2011, 73-80. · Zbl 1236.54004 |

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