= 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>,
$$
\boxed{U=\mathcal M}.
$$
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