## 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.

### 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

### Keywords:

density point; density topology

### Citations:

Zbl 1236.54004; Zbl 1222.26005; Zbl 1135.28001; Zbl 1151.54005
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.