These topologies are not Hausdorff. However, given a filter F over N by an elementary construction it is easy to define a Hausdorff topology of the same complexity as the filter F .
Open topologies are necesarilly clopen.
References F. Van Engelen. On Borel ideals. A. S. Kechris. Rigidity properties of Borel ideals on the integers. Pre-print , 1996.