Interior in the simple density topology (Q952596)

The density topology on \(\mathbb{R}\), the real line, is defined in terms of points of density. It is known that a point \(x\) in \(\mathbb{R}\) is a density point of a Lebesgue measurable subset \(A\) of \(\mathbb{R}\) if and only if the sequence \(\{\chi_{n \cdot (A-x)\cap[-1,1]}\}_{n\in\mathbb{N}}\) of characteristic functions converges in measure to \(\chi_{[-1,1]}\). The authors modified this requirement of convergence in measure to pointwise convergence almost everywhere [Rend. Circ. Mat. Palermo (2) 53, No. 3, 344--352 (2004; Zbl 1194.26002)] in order to define the simple density topology \(\tau_s\) on \(\mathbb{R}\). This topology is strictly stronger than the natural topology on \(\mathbb{R}\) and weaker than the density topology on \(\mathbb{R}\). In the present paper, in analogy to the corresponding results for the density topology, for each subset \(A\) of \(\mathbb{R}\), the authors determine the interior of \(A\) in \(\tau_s\) in terms of an operator \(\Phi_{s}^{\alpha}\), where the operator \(\Phi_{s}^{\alpha}\) for a countable ordinal \(\alpha\) is defined as \(\Phi_{s}^{\alpha}(E)=\Phi(\Phi_{s}^{\beta}(E))\) if \(\alpha=\beta+1\) and \(\Phi_{s}^{\alpha}(E)=\bigcap_{\beta<\alpha}\Phi_{s}^{\beta}(E)\) if \(\alpha\) is a limit ordinal for a Lebesgue measurable subset \(E\) of \(\mathbb{R}\). They prove that the interior of \(A\) in \(\tau_s\) is given by the formula \(\text{Int}_s(A)=A\cap \Phi_{s}^{\alpha}(B)\) where \(B\) is any measurable kernel of \(A\) and \(\beta\) is the smallest countable ordinal for which \(\Phi_{s}^{\beta}(B)=\Phi_{s}^{\beta+1}(B)\). This paper shows a good interplay between general topology and measure theory.