Omega-automata accept or reject infinite words. A word is accepted by an existential nondeterministic automaton if the word has an infinite run starting in initial states and satisfying the acceptance condition of the automaton.
The acceptance condition of the basic omega-automata is defined with a subset of the states which are called the accepting states.
For instance, a word is accepted by a safety automaton if it has a run starting in initial states and visiting only the accepting states of the automaton. A word is accepted by a liveness automaton if it has a run starting in initial states and visiting at least once an accepting state of the automaton.
For safety automata, the "safe" states are just a synonym for its accepting states, so they are given and define the automaton itself.
