Let P_{i \in I} be a (possibly infinite) family of safety properties and let P = \bigcap _{i \in I} P_i. Take any behaviour \sigma \notin P. Then there exists some j \in I such that \sigma \notin P_j. Since P_j is a safety property, there is a finite prefix u of \sigma such that no extension of u lies in P_j. But then no extension of u can lie in \bigcap _{i \in I} P_i, because membership in the intersection requires membership in P_j, which is already ruled out. Hence u is a bad prefix for P, so P is a safety property.