Revisiting Variations in Topological Transitivity

Topological dynamical systems $(X,T)$ are actions $T \times X \to X$, given as $(t, x) \to tx$, on a compact, Hausdorff topological space $X$ with $T$ as an acting group or monoid. We take up the property of topological transitivity especially for semiflows $(X,S)$ and discuss the variations in its definitions.