Porosity, \({\mathcal I}\)-density topology and abstract density topologies
In 1982 the reviewer has introduced a category analogue of the density topology [Real Anal. Exch. 8(1982/83), 16-20 (1983)]. The definition was based upon the operator of lower density, the definition of which uses the algebraic structure of the real line. The author has proved that the notion of porosity and basic topological notions are sufficient to describe the above mentioned \({\mathcal I}\)-density topology. Because of the porosity is strictly connected with the metric, the \({\mathcal I}\)-density topology can be generalized to arbitrary metric space. The basic properties of such topologies (denoted by \(p^*\)-topology) are studied. Among others it is proved that any real function which is continuous with respect to \(p^*\)-topology satisfying the essential radius condition is of first Baire class with respect to the topology generated by the metric p. Also a.e. and r. modifications of category density topologies are considered. At the end there are some remarks concerning ``canonical'' density topologies and some related open problems.
category analogue of the density topology
scientific article; zbMATH DE number 4044117