 Solution

Write $I^{+}(U)=U=I^{-}(U)$. Since every chronological future and chronological past is open, $U$ is an open set. We show its complement is also open.

Take $p\notin U$ and choose $p_{-}\ll p\ll p_{+}$ in a convex normal neighborhood. The diamond $V=I^{+}(p_{-})\cap I^{-}(p_{+})$ is a neighborhood of $p$. If $x\in V\cap U$, then $p_{-}\ll x$ and $U=I^{-}(U)$ imply $p_{-}\in U$; then $p_{-}\ll p$ and $U=I^{+}(U)$ imply $p\in U$, a contradiction. Thus $V$ lies in the complement of $U$.

Hence $U$ is nonempty, open, and closed. Since $\mathcal{M}$ is a connected space,