Counterexample. Safety closure does not distribute over intersection [ltp-000S]

Take the properties P = \{ \sigma \mid \texttt {a}\ \text {occurs infinitely often in}\ \sigma \} and R = \{ \sigma \mid \texttt {a}\ \text {occurs finitely many times in}\ \sigma \}. Both have a safety closure of \Sigma ^\omega , i.e. \overline {P} = \overline {R} = \Sigma ^\omega , however P \cap R = \emptyset so the safety closure of the intersection is \emptyset , not \Sigma ^\omega .