Generalized Büchi Automata - Formal definition of a state appearing infinitely often?
The definition and example are both correct.

If the automaton reads&nbsp;$a$ in state&nbsp;$2$ or reads&nbsp;$b$ in state&nbsp;$1$, then it rejects, because it has no state to go to.  So, if the machine reads an infinite amount of its input, then it must read&nbsp;$a$ every time it is in&nbsp;$1$ and read&nbsp;$b$ every time it is in&nbsp;$2$.  If the machine reads&nbsp;$b$ only a finite number of times, it only enters&nbsp;$2$ a finite number of times, so it must reject. Therefore, any string that's accepted must contain an infinite number of&nbsp;$b$s.  Since $b$s can only be read in state&nbsp;$2$ and the only way of getting there is to read an&nbsp;$a$ from state&nbsp;$1$, any accepted string must contain an infinite number of&nbsp;$ab$s.