Compact group acting on regular space

Question

Let $G$ be a compact topological group, $X$ be a regular topological space. Then the quotient space given by the continuos action of $G$ on $X$, $X/G$ is also regular.
Here's my attempt, though I feel it's wrong for some reason:
Let $\bar{y} = [y] \in X/G, \bar{y}\not\in F\subset X/G$ closed. Also, let $p:X\rightarrow X/G$ be the projection. Then: $$(p^{-1}(\bar{y}):=G\cdot y)\cap (p^{-1}(F):=G\cdot F)=\emptyset$$ Clearly, $G\cdot F$ is closed since $p$ is continuous. Since $X$ is regular, then, for each $z\in G\cdot y$, $\exists U_z$ neighbourhood of $z$, $V_z$ open set containing $G\cdot F$ such that $U_z\cap V_z=\emptyset$. Thus, $$G\cdot y\subset \bigcup_{z\in G\cdot y}U_z$$ Now, for each $y\in X$, define $m_y:G\rightarrow X$, by $m_y(g)=g\cdot y$. Clearly, $m_y$ is continuous for every $y$, and $m_y(G)=G\cdot y$. It follows that $\{m_y^{-1}(U_z)\}_z$ is an open cover for $G$, and, aince it is compact, admits a finite subcover $$G=\bigcup_{i=1}^nm_y^{-1}(U_{z_i})$$ From here it's pretty straightforward: It follows that $$G\cdot y\subset \bigcup_{i=1}^nU_{z_i}=U$$ And then $$G\cdot F\subset\bigcap_{i=1}^nV_{z_i}=V$$ So that, since $p$ is an open map and surjective, $p(U),\,p(V)$ are open subsets of $X/G$ such that $\bar{y}\in p(U),\,F\subset p(V),\,p(U)\cap p(V)=\emptyset$.
Are there any mistakes in this proof?

Answer 1

If I understood your notation right, you didn’t prove the key moment: why $p(U)\cap p(V)=\varnothing$.

As minor remarks I note that there is no need to take preimages $m_y{-1}(U_z)$, it suffices to use compactness of a set $G\cdot y$. It seems you tend to consider $G\cdot F$ as a subset of $X$, whereas formally $G$ is a set of equivalence classes, each of which is a subset of $X$. The map $p$ , I guess, is called a quotient map, not a projection. Finally, I provide references to general facts implying the required claim.

References

[165] Ryszard Engelking, General Topology (PWN, Polsih. Scientific. Publ., Warszawa), 1977.

[AT] Alexander V. Arhangel'skii, Mikhail G. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.

[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.