Definition 40.1.5 (Transitive).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is topologically transitive with respect to $G$ if $Gx$ is dense in $X$.

The flow $G \curvearrowright X$ is point transitive if it admits a topologically transitive point.

Finally, $G \curvearrowright X$ is topologically transitive if for any $U, V \subset X$ non-empty and open, there exists $g \in G$ such that $gU \cap V \ne \emptyset$^[1].