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

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Last modified May 1, 1999.