Let $G$ be a topological group acting continuously on an Hausdorff space $X$. We say that the action is topologically transitive if for any open $U\subseteq X$, $UG$ is dense in $X$, and we say that the action is point transitive if there exist a point $x$ with dense orbit $xG$.
It's straightforward to see that point transitive implies topologically transitive, and exercise 6 in the first chapter of Auslander's "Minimal Flows and their Extensions" asks to prove that the converse also holds if $X$ is second countable.
It's easy to see that if $Y\subseteq X$ is the set of transitive points (points with a dense orbit), then $Y=\bigcap UG$, where $U$ varies over nonempty sets in a basis for the topology of $X$, and so if $X$ is second countable there is a comeager $G_\delta$ set of points with dense orbits. With some extra assumptions on $X$, such as $X$ compact, or more generally $X$ Baire, this is enough to solve the exercise I mentioned, but I don't see why this intersection must be nonempty in general.
The book so far has been very careful in stating explicitely when $X$ is assumed compact, so I'm not sure whether this is a typo in the exercise or whether I'm missing something, the same result is also found in "Topological Dynamics" by Gottschalk and Hedlund as theorem 9.20, but with even stronger assumptions, namely they ask for the phase space to be Polish (hence Baire).
So in conclusion my doubt is the question in the title: Suppose that the topological group $G$ acts continuously on the second countable, Hausdorff space $X$ with an action that is topologically transitive. Must the action be point transitive? What is a counterexample if the answer is negative?
Best Answer
Take the natural $\Bbb Z$-action on $\{0, 1\}^{\Bbb Z}$ by shift, and consider the subspace $X$ of periodic $2$-sided sequences of some unspecified period length. Clearly the original action is topologically transitive, and $X \subset \{0, 1\}^{\Bbb Z}$ is dense. Thus, the $\Bbb Z$-action on $X$ is in particular topologically transitive as well. Every orbit is periodic.