# topological dynamics – A reference to the fact that a topologically transitive action of a group on a compact metric space has a dense orbit

I need an appropriate reference to the following obvious fact:

A group action $$G$$ on a compact metrable space $$K$$ is topologically transitive (= the orbit of any non-empty open set is dense) if and only if it is the orbit $$Gx$$ from a certain point $$x in K$$ is dense in $$K$$.

For a cyclic group, this characterization is proven here. I hope that some textbooks on topological dynamics should contain such a basic fact. Help me!