Point transitivity
= Point transitivity
{title2=$\overline{\{f^n(x):n\geq0\}}=X$}
= Point-transitive
{synonym}
A <continuous map> on a nonempty <topological space> is point-transitive if it has a <dense> forward <orbit>. This is one convention for <topological transitivity>, but on spaces with isolated points it must be distinguished from the open-set intersection definition. On a nonempty <compact metric space>, the positive-time open-set intersection property implies point transitivity: for each member $U_j$ of a countable <basis of a topology>, $\bigcup_{n\geq1}f^{-n}(U_j)$ is open and dense, and the <Baire category theorem> supplies a point in their intersection. Conversely, a <surjective> point-transitive <continuous map> has every tail of its dense orbit dense, since $f^k(X)=X$ and continuity sends its orbit closure into the closure of that tail. The map $f(1/j)=1/(j+1)$, $f(0)=0$ on $\{0\}\cup\{1/j:j\geq1\}$ is point-transitive but fails the open-set intersection property.