I have that X is a compact topological space and G is a group that acts on X minimally in the sense that $\forall$ x $\in$X, $\overline{\cup_{g\in G} (g.x)}$ = X.
I want to show that given any non-empty open subset U of X, I can find finitely many elements g$_{1}$, g$_{2}$, ... , g$_{n}$ of G such that $\cup_{i=1}^{n}$ (g.U) = X.
It seems to me that since orbit of any element comes arbitrarily close to any other element, somehow orbit of any open set should come arbitrarily close to any open set. If I could show that orbit of open set U actually forms an open set (may be around every point), then I could invoke compactness. However, that seems as if I am trying to draw a stronger conclusion, in which case, I have no idea how to approach this problem.