Refinements of the density and ℐ-density topologies

Abstract

Given an arbitrary ideal J \mathcal {J} on the real numbers, two topologies are defined that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of J \mathcal {J} . This shows why the restriction to Baire sets is necessary in the usual definition of the J \mathcal {J} -density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal.