Counterexample. Safety properties are not closed under infinite union [ltp-000E]

Consider the family of properties P_{i \in \mathbb {N}} = \{ \sigma \in \Sigma ^\omega \mid \sigma _i = \texttt {a}\}, i.e. the property where the ith state is \texttt {a}. Each P_i is a safety property, as all violating behaviours are extensions of bad prefixes of length i. Their union \bigcup _{i \in \mathbb {N}} P_i, however, is not a safety property, as any finite prefix can be extended to a good prefix by appending an \texttt {a}-state, and therefore cannot be a bad prefix.