I'm trying to solve this problem and I don't know how to start
Let $M$ be a connected time-oriented Lorentz manifold of dimension $n$. Let $$J^{+}(K)=\{q\in M: \text{there is a $p\in K$ with $p\leq q$}\}.$$ $p\leq q$ means that there exist a causal curve from $p$ to $q$. Suppose that for every $p\in M$, the sets $J^{+}(p)$ and $J^-(p)$ are closed. Prove that if $K$ is compact, then $J^{+}(K)$ is closed.
I tried to prove that a convergent sequence un $J^{+}(K)$ converge to a point in $J^+(K)$, but nothing...
My last comment was not very accurate.
Edit:
Actually my comment is not so useful ... I think that the argument goes in this way: Consider $q \in \overline{J^{+}(K)}$, so, for any $w \in I^{+}(q)$ we have that $I^{-}(w) \cap J^{+}(K) \neq \emptyset$. Therefore by taking a sequence $\{q_{n}\} \subset I^{+}(q)$ such that $q_{n+1} \ll q_{n}$ and $q_{n} \rightarrow q$. We can consider the following subset of $K$: $K \cap J^{-}(q_{n})$ which will not be empty and nested, so, $\cap_{n \in N}(K \cap J^{-}(q_{n})) \neq \emptyset$ and use a point $p$ of this set to show that $q \in J^{+}(p) \subset J^{+}(K)$. There are some hidden steps in here, but if you work it out it will be a good exercise.