Let $G$ be a countable group and $(X, d)$ be a compact metric space and $\varphi:G\times X\to X$ is a group action.
I am trying to show that :
For $R\subseteq G$ there is $x\in X$ with $\inf_{g\in R}(d(\varphi(g, x), x)=0$ if and only if for every open cover $\{U_1, U_2, \ldots, U_r\}$ of $X$, there exist $1\leq j\leq r$ and $g\in R$ such that $\varphi(g, U_j)\cap U_j\neq\emptyset$
It is easy to see that if for $R\subseteq G$ there is $x\in X$ with $\inf_{g\in R}(d(\varphi(g, x), x)=0$, then for every open cover $\{U_1, U_2, \ldots, U_r\}$ of $X$, there exist $1\leq j\leq r$ and $g\in R$ such that $\varphi(g, U_j)\cap U_j\neq\emptyset$.
Question. The converse is true?