Refinements of the density and $\mathcal {I}$-density topologies

Abstract:

Given an arbitrary ideal $\mathcal {J}$ 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 $\mathcal {J}$. This shows why the restriction to Baire sets is necessary in the usual definition of the $\mathcal {J}$-density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal.