  Given a topological $G$-space we consider equations with parameters over $G$.
In particular we formulate some very general conditions on words with
parameters $w(\bar{y},\bar{g})$ over $G$ which guarantee that the inequality
$w(\bar{y},\bar{g})\neq 1$ has a solution in $G$. These results are illustrated
in some typical situations, in particular standard actions of Thompson's group
$F$ and branch groups are considered.
  The major results of this paper appeared in some form in Section 2 of the PhD
thesis of the second author (avalable at arXiv:1308.6330).
