Pointwise density topology with respect to admissible \(\sigma \)-algebras (Q2843893)

The paper considers the notion of p-density and density-type topologies defined using this notion. Zero is a p-density point of a set \(A\subset \mathbb {R}\) if the sequence \(\left \{\chi _{nA\cap \left [-1,1\right]}\right \}_{n\in \mathbb {N}}\) converges to the function \(\chi _{\left [-1,1\right]}\); \(x\) is a p-density point of \(A\)\ if \(0\) is a p-density point of \(A-x\). The family \(\tau_{p}\), consisting of sets for which any point is a p-density point, forms a topology.NEWLINENEWLINEThe authors define ``admissible'' \(\sigma \)-algebras \(S\subset 2^{\mathbb {R}}\)-such that \(\tau_{p}\cap S\) form a topology (for example, the \(\sigma \)-algebra \(\mathcal {L}\), consisting of Lebesque measurable sets and the \(\sigma \)-algebra \(\mathcal {B}_{a}\), consisting of sets having the Baire property, are admissible, but the \(\sigma \)-algebra of Borel sets is not). They present some properties of the topologies \(\tau_{p}\cap S\) and compare them with other density-type topologies.