Counterexample. Liveness properties are not closed under intersection [ltp-000U]

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 are liveness properties, i.e. dense (\overline {P} = \overline {R} = \Sigma ^\omega ), however P \cap R = \emptyset which is not a liveness property (as \overline {\emptyset } = \emptyset not \Sigma ^\omega ).