Wagner-Bojakowska, Elżbieta; Wilczyński, Władysław The interior operation in a \(\psi\)-density topology. (English) Zbl 0949.26002 Rend. Circ. Mat. Palermo, II. Ser. 49, No. 1, 5-26 (2000). Let \(\psi :(0,\infty)\to (0,\infty)\) be a continuous non-decreasing function such that \(\lim_{x\rightarrow 0^+}\psi (x)=0\). A point \(x \in {\mathcal R}\) is a \(\psi \)-density point of a measurable set \(A \subset {\mathcal R}\) if \(\lim_{h \rightarrow 0^+}\frac{m(A'\cap [x-h,x+h])}{2h\psi (2h)} = 0\). The family of all measurable sets such that each point \(x \in A\) is a \(\psi \)-density point of \(A\) is a topology called \(\psi \)-topology. In this paper it is proved that for an arbitrary set \(A\) its interior in a \(\psi \)-topology is equal to \(A\cap \varphi ^{\beta }(B)\), where \(B\) is a measurable kernel of \(A\) and \(\beta \) is some countable ordinal. Moreover, each countable ordinal \(\beta \geq 1\) realizes the interior of some measurable set \(A\). Reviewer: Zbigniew Grande (Bydgoszcz) Cited in 4 Documents MSC: 26A03 Foundations: limits and generalizations, elementary topology of the line 28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets Keywords:\(\psi \)-topology; \(\psi \)-density point; order of a measurable set PDFBibTeX XMLCite \textit{E. Wagner-Bojakowska} and \textit{W. Wilczyński}, Rend. Circ. Mat. Palermo (2) 49, No. 1, 5--26 (2000; Zbl 0949.26002) Full Text: DOI References: [1] Oxtoby, J. C., Measure and category (1971), New York: Springer Verlag, New York · Zbl 0217.09201 [2] Ostaszewski, K., Continuity in the density topology, Real Analysis Exchange, 7, 2, 259-270 (1981) · Zbl 0494.26004 [3] Taylor, S. J., On strengthening the Lebesgue Density Theorem, Fund. Math., 46, 305-315 (1959) · Zbl 0086.04601 [4] Terepeta, M.; Wagner-Bojakowska, E., ψ-density topologies, Rend. Circ. Mat. Palermo, 48, 451-476 (1999) · Zbl 0963.26003 · doi:10.1007/BF02844336 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.