Group action on a completely regular space I am trying to prove that $X/G$ is completely regular provided that $X$ is a completely regular space and $G$ is a compact hausdorff group acting on $X$.
Let $p: X \rightarrow X/G$ be the canonical quotient map. So I choose a point $\overline{x} \in X/G$ and $C \subseteq X/G$ closed subset with $\overline{x}\notin C$. Then I know that $D = p^{-1}(C)$ is closed in $X$ and it is disjoint to $G\cdot x = p^{-1}(\overline{x})$. Therefore, for any $g\in G$, there is a continuous function $f_g:X \rightarrow \mathbb{R}$ satisfying $f_g(gx) = 0$ and $f_g(D)= 1$. 
I would like to use the compactness of $G$ to be able to consider just finite of such functions, but I don't really know what else to do. I was also trying to construct an $G$-invariant map $f:X \rightarrow \mathbb{R}$ satisfying $f(x) = 0$ and $f(D)= 1$ and thus conclude that the map factor uniquely throughout the quotient.

I recall the following known facts 
From [AT]:

I guess [165, Theorem 1.4.13] is the same as [Eng, Theorem 1.4.13]. 
From [Eng]:


References
[165] Ryszard Engelking General Topology, PWN, Polish Scientific Publ., 1977.
[AT] A.V. Arhangel'skii, M. Tkachenko Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.
[Chaber 1972] Chaber, J. Remarks on open-closed mappings, Fund. Math. 74 (1972), 197-208.
[Eng]  Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
[Frolík 1961] Frolík Z. Applications of complete familes of continuous functions to the theory of $Q$-spaces, Czech. Math. J 11 (1961),  115-133.
[Ponomarev 1959] Ponomarev V.I. Open mappings of normal spaces,
Dokl. Akad. Nauk SSSR 126 (1959), 716-718, (in Russian).
