Sharper changes in topologies
  Becker and Kechris showed that if a Polish group G acts continuously on a Polish space X, then for any invariant Borel set B we can change the topology on X so that B becomes open, the Borel structure is preserved, and the action continues to be continuous.   In this brief paper a short proof is presented for their theorem. The method also gives optimal bounds in terms of minimizing the change to the original topology. 
Greg Hjorth