Refinements of the Density and ^-density Topologies

Given an arbitrary ideal f on the real numbers, two topologies are denned that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of f . This shows why the restriction to Baire sets is necessary in the usual definition of the ./"-density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal. In studying real functions it is often helpful to endow the real numbers R with a topology finer than the natural one, denoted by ST&, arising from its order. The most common examples of such topologies are the density topology Hjr and its category analog, the J" -density topology ^> . To recall the definitions of these topologies we need the following notions. A point jc € R is said to be a density point of A c R if (1) m(An(x-l/n,x+l/n)) = v ' «-oo 2/n