= Solution
The convention intended by this question is \b[existence of a dense forward orbit], also called <point transitivity>. On a nonempty <compact metric space>, this means that some $x\in X$ satisfies
$$
\boxed{\overline{\{f^n(x):n\geq0\}}=X.}
$$
Equivalently, that forward <orbit> visits every nonempty <open set>.
The other common definition of <topological transitivity> uses intersections of iterated <open sets>, as in part (b). The distinction matters on spaces with <isolated points>: parts (b) and (c) ask precisely how the two conventions are related. We use <point transitivity> for the dense-orbit convention throughout this solution. As usual for this definition, the <state space> is assumed nonempty; on an empty space the open-set condition is vacuous and there is no point with a forward <orbit>.
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