For an analytic space $U$ equipped with an action of a group $\Gamma$, call a subset $Z\subseteq U$ $\Gamma$-closed iff it is a closed analytic subset and each of its irreducible components is an irreducible component of a $\Gamma$-invariant closed analytic set.
When do $\Gamma$-closed sets form a topology? Does it have a name and are there any references discussing it ?
What I want to know is the following: Under what conditions the following is true:
(i) $\Gamma$-closed sets form a topology. (ii) for an analytic map $f:U\rightarrow U'$, the map $f:U/\Gamma \rightarrow U'/\Gamma'$ being closed and well-defined implies $f_*:U \rightarrow U'$ is closed as well in the topology of $\Gamma$- and $\Gamma'$-closed sets ?
For (i), it is enough to prove that the intersection of two $\Gamma$-closed analytically irreducible closed sets is $\Gamma$-closed.
I am mostly interested in the case when $\Gamma$ is the topological fundamental group of a complex algebraic variety acting on its univeresal covering space $U_A$. Informally, the idea is to define a somewhat "finer" variant of the etale topology ( induced on the inverse limit of finite etale covers of $A(C)$.) For $\Gamma$ the algebraic fundamental group and $U$ the inverse limit as above, both (i) and (ii) are easy.
There is a proof of this under assumption that $U$ and $U'$ are universal covering spaces of projective algebraic varieties with subgroup separable (aka lerf) fundamental groups, by a rather ugly technical argument and not quite in this form ( reference, p.11 and p.20). When $A(C)$ is a group, probably it also follows from a purely algebraic proof in a model theory paper (which also generalised to prime characteristic).