Continuous action Let $G$ be a polish group, $H$ an open subgroup of $G$. Now assume that $H$ acts by isometries (For all $h\in H$, the map $X\ni x\longmapsto X$ is an isometry) and continously on a metric space $(X,\delta)$. We define
$$F=\{f:G\longrightarrow X|\,\forall h\in H,\,\,\forall g\in G,\,\,f(gh)=h^{-1}f(g)\}$$
$G$ act on $F$ by left translation $$G\times F\ni (g,f)\longmapsto \tau_{g}f \in F$$ where $\tau_{g}f(x)=f(g^{-1}x)$ for all $x\in G$.
My goal is to show that the previous action of $G$ on $F$ is continuous
We assume that we can equipped $F$ with the following metric:
$$d:F\times F\ni(f,h)\longmapsto d(f,h)=\underset{g\in G}{\overset{}{\sup}}\,\delta(f(g),h(g))\in \mathbb{R}_{+}$$
If $F$ is equiped with the previuous action, then the action of $G$ on $F$ is by isometries
Since $H$ is open in $G$, we only need to show that the action of $H$ on $F$ is continuous. How to show that this fact? where $F$ is  equipped by the toplogy induced by the previous metric.
Thank for any help
 A: This question is my own research. It is not homework. 
Thank for your previous answer. For this question, i have an idea. But i am not sure that i am right:
In fact, since the action is by isometries, the map $F\in f\longmapsto \tau_{g}f\in F$ is continuous for all $g\in G$. Now fix $f\in F$ and suppose that $g_{n}\longrightarrow g$ in $G$, then $g^{-1}_{n}a\longrightarrow g^{-1}a$ for all $a\in G$. Therefore, $f(g^{-1}_{n}a)\longrightarrow f(g^{-1}a)$ since $f$ is continuous. So we have $\tau_{g_{n}}f\longrightarrow \tau_{g}f$ pointwise. Now since $G$ is separable and $H$ is open, then the quotient space is countable(separable+discrete). Therefore the following $\underset{a\in G}{\overset{}{\sup}}\,\delta(f(g_{n}^{-1}a),f(g^{-1}a))$ is in fact a max. Therefore, We have:
      $$d(\tau_{g_{n}}f,\tau_{g}f)=\underset{a\in G}{\overset{}{\max}}\,\delta(f(g_{n}^{-1}a),f(g^{-1}a))\longrightarrow 0$$
Thank for givin your opinion about this idea.
A: More precisely, since $\underset{g\in G}{\overset{}{\sup}}\,\delta(f(g),h(g))$ can be infinite this map failed two be a metric on $F$ in general.\
In this case i can consider the following equivalence relation on $F$:
$$f\sim g \,\Longleftrightarrow d(f,g)< \infty$$
If $F_0$ is any class for the relation $\sim$, then $F_0$ is invariant under the action of $G$.
In fact, my goal is to prove that the action on any class $F_0$ is continuous.
Thank one more for your help.
