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.

**Last modified May 1, 1999.**