# Are both safety and liveness properties closed under finite intersection?

Given two safety properties $$P_1$$ and $$P_2$$, is their intersection $$P_1 \cap P_2$$ also a safety property? Similarly, if $$P_1$$ and $$P_2$$ are liveness properties, is $$P_1 \cap P_2$$ also a liveness property?

Liveness properties as defined by Alpern and Schneider are dense. They are not closed under intersection as soon as there are two elements in the state space. Consider the two-element state space $$\{0,1\}$$. For state $$i$$ define $$a_i$$ as the set of infinite sequences of states that have a suffix consisting of $$i$$ only. Each $$a_i$$ is a liveness property but $$a_0 \cap a_1 = \emptyset$$ cannot be a liveness property because it's not dense.