I need a proper reference to the following obvious fact:



> 
> An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $U$ is dense) if and only if it the orbit $Gx$ of some point $x\in K$ is dense in $K$.
> 
> 
> 


For a cyclic group this characterization is proved [here](https://www.researchgate.net/publication/251140179_Existence_of_a_dense_orbit_and_topological_transitivity_When_are_they_equivalent). I hope that some textbook in topological dynamics should contain such a basic fact. 



A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit
See Theorem 9.20 of "Topological Dynamics" by Gottschalk and Hedlund. It states that, for systems $(X,G)$ whose phase space is non-empty complete separable metric, point transitivity (a point having a dense orbit) and topological transitivity (every non-empty open set having dense orbit) are equivalent.


