The interior operation in a \(\psi\)-density topology
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 \(A\).
scientific article; zbMATH DE number 1462203