Sharper changes in topologies

Abstract:

Let $G$ be a Polish group, $\tau$ a Polish topology on a space $X$, $G$ acting continuously on $(X,\tau )$, with $B\subset X$ $G$-invariant and in the Borel algebra generated by $\tau$. Then there is a larger Polish topology $\tau ^*\supset \tau$ on $X$ so that $B$ is open with respect to $\tau ^*$, $G$ still acts continuously on $(X,\tau ^*)$, and $\tau ^*$ has a basis consisting of sets that are of the same Borel rank as $B$ relative to $\tau$.