Counterexample. Decision properties are not closed under infinite union nor intersection [ltp-0010]

Follows from Counterexample [ltp-000E]. Each P_i is both a safety property and a guarantee property. Therefore, each P_i is a decision property. But the safety closure of their infinite union \bigcup _{i \in \mathbb {N}} P_i is the strictly larger set \Sigma ^\omega . Therefore, their infinite union is not even a safety property so certainly not a decision property.

Therefore, decision properties are not closed under infinite union. Because decision properties are closed under complement, they cannot be closed under infinite intersection either.