Closure of the exterior Cauchy section meets only the double horizon

The sets $I_M^{\pm }(U)$ are unions of open chronological futures or pasts of points of $U$, so $E$ is open. It is also causally convex for this arbitrary $U$: if a future causal curve in $M$ has endpoints $a,b\in E$ and passes through $z$, choose $u,v\in U$ with $u{\ll }_{M}a$ and $b{\ll }_{M}v$. Its restrictions give $a{\le }_{M}z{\le }_{M}b$, and push-up gives $u{\ll }_{M}z{\ll }_{M}v$. Thus every point of the curve lies in $E$. Consequently any ambient causal or timelike curve between points of $E$ is also a curve in $E$. No connectedness of $U$ is used.

By [F4], $C$ is acausal in $E$. Suppose $p\in {\bar{C}}^E\setminus C$. A small coordinate straight segment through $p$, in a future timelike direction, is smooth timelike and lies in $E$: continuity of the metric and time orientation preserves its timelike direction after shrinking the coordinate interval. By [F4] it lies on an inextendible causal curve in $E$. That curve meets $C$ at some $x\mathrm{\ne }p$. The subarc between $x$ and $p$ contains a nondegenerate part of the timelike segment next to $p$; [F3] therefore gives $x{\ll }_{E}p$ or $p{\ll }_{E}x$. Openness of chronology in $E$ gives a neighborhood of $p$, chosen to omit $x$, whose points have the same relation to $x$. It meets $C$ by the closure assumption, contradicting acausality. Thus $C$ is closed in $E$. Since $E$ is open, every $p\in {\bar{C}}^M\setminus C$ lies outside $E$.

Fix $p\in {\bar{C}}^M\setminus C$, and suppose $p\in I_M^{+}(U)$. Choose a future timelike curve $\gamma :[0,1]\to M$ from some $u\in U$ to $p$. For $0<t<1$, put $z=\gamma (t)$. Then $u{\ll }_{M}z{\ll }_{M}p$. The open set $I_M^{+}(z)$ contains $p$ and hence meets $C$ at some $c$. Since $c\in E\subset I_M^{-}(U)$, there is $v\in U$ with $c{\ll }_{M}v$. Concatenation gives $z{\ll }_{M}c{\ll }_{M}v$, so $z\in I_M^{-}(U)$; also $u{\ll }_{M}z$ gives $z\in I_M^{+}(U)$. Therefore $\gamma ((0,1))\subset E$.

Choose $0<s_0<s<1$. Apply [F4] in $E$ to the compact segment $\gamma {\mathrm{\mid }}_{[s_0,s]}$. Take the resulting inextendible causal curve's entire past portion up to its parameter corresponding to $\gamma (s)$, and then append $\gamma {\mathrm{\mid }}_{[s,1)}$, matching the junction parameters by a positive affine change. This concatenation is a future causal curve in $E$ by [F5]. Its past end has no limit in $E$, since it is unchanged from the inextendible extension. Its future end converges in $M$ to $p\mathrm{\notin }E$, and so cannot converge to a point of $E$, by uniqueness of limits in the Hausdorff manifold $M$. It is therefore inextendible in $E$. This construction needs only the segment-extension assertion, not an assertion that the entire extension is timelike.

The Cauchy property gives a point $x\in C$ on the curve of step 4.1. Choose a point $z$ on its appended timelike tail strictly after that intersection. Then $x{\le }_{M}z{\ll }_{M}p$, so push-up gives $x{\ll }_{M}p$. The open neighborhood $I_M^{+}(x)$ of $p$, intersected with $M\setminus \{x\}$, meets $C$ at some $q\mathrm{\ne }x$ because $p\in {\bar{C}}^M$ and $p\mathrm{\ne }x$. Thus $x{\ll }_{M}q$. The connecting timelike curve stays in $E$ by step 1.1, yielding $x{\ll }_{E}q$, contrary to the acausality in step 2.1. Hence $p\mathrm{\notin }{I}_M^{+}(U)$.

Apply steps 3.1--5.1 with time orientation reversed. The exterior $E$ and Cauchy condition are unchanged, while $I_M^{+}(U)$ and $I_M^{-}(U)$ are interchanged, so $p\mathrm{\notin }{I}_M^{-}(U)$ as well. Finally $C\subset E\subset I_M^{\pm }(U)$ implies $p\in {\overline{I_M^{+}(U)}}^M\cap {\overline{I_M^{-}(U)}}^M$. Both chronological sets are open; exclusion from each therefore places $p$ in both ambient boundaries. This proves the claimed inclusion, together with the relative closedness already proved.