A Cauchy section cannot end on only one chronological horizon

Any future causal segment in $E$ with both endpoint values in $C$ would, by clause (c) in [F4], lie on an inextendible causal curve in $E$ having two distinct parameter values in $C$. This contradicts the Cauchy condition, even if the endpoint values coincide. In particular $C$ is acausal in $E$. An ambient timelike curve in $D$ whose endpoints lie in $C$ lies wholly in $E$ by causal convexity, so no such timelike curve exists either. Thus $C$ is achronal also for the ambient chronological relation.

Let $p\in {\mathrm{\partial }}_{\mathrm{\Sigma }}C$. Since $E$ is open and $C=\mathrm{\Sigma }\cap E$, the set $C$ is open in $\mathrm{\Sigma }$ and its boundary is disjoint from $C$. Hence $p\mathrm{\notin }E$. Every neighborhood of $p$ in $D$ meets $C\subset A\cap B$, so $p\in \bar{A}\cap \bar{B}$. As $A$ and $B$ are open and $p\mathrm{\notin }A\cap B$, at least one of $p\in \mathrm{\partial }A$ and $p\in \mathrm{\partial }B$ holds.

Suppose $p\in \mathrm{\partial }A\setminus \mathrm{\partial }B$. Then $p\in B$ and $p\mathrm{\notin }A$. In $D$, take the geodesic $\rho$ with $\rho (0)=p$ and ${\rho }^{\mathrm{\prime }}(0)=N_p$. By [F3] it is smooth and future timelike. Since $B$ is open, continuity permits $\epsilon >0$ such that $\rho$ is defined on $[-\epsilon ,0]$ and this segment lies in $B$. For $-\epsilon \le t<0$, put $z=\rho (t)$; its timelike segment to $p$ gives $z{\ll }_{D}p$. Openness makes $I_D^{+}(z)$ a neighborhood of $p$, which therefore meets $A$ because $p\in \bar{A}$. Choose $w\in A\cap I_D^{+}(z)$; the past-set property of $A$ gives $z\in A$. Thus $\rho ([-\epsilon ,0))\subset E$, and its future limit $p$ is outside $E$.

Define $\gamma :[0,\mathrm{\infty })\to E$ by $\gamma (s)=\rho (-\epsilon e^{-s})$. This is smooth with nonzero future timelike velocity, since the parameter derivative is $\epsilon e^{-s}>0$. It tends to $p$ in $D$ as $s\to \mathrm{\infty }$; it cannot converge to a point of $E$, by uniqueness of limits in the Hausdorff manifold $D$ and $p\mathrm{\notin }E$. Thus it is future-inextendible in $E$. Apply [F5] to obtain an inextendible future causal ${\gamma }^{\ast }:I^{\ast }\to E$ agreeing with $\gamma$ on $[0,\mathrm{\infty })$. The Cauchy property gives $a\in I^{\ast }$ with $c={\gamma }^{\ast }(a)\in C$. Choose $s>\max \{0,a\}$. The segment of ${\gamma }^{\ast }$ gives $c{\le }_E\gamma (s)$, while the remaining timelike geodesic segment gives $\gamma (s){\ll }_{D}p$. Push-up yields $c{\ll }_{D}p$. Now $I_D^{+}(c)$ is an open neighborhood of $p$ and $p\in {\mathrm{\partial }}_{\mathrm{\Sigma }}C$, so it contains some $c^{\mathrm{\prime }}\in C$. This yields $c{\ll }_D{c}^{\mathrm{\prime }}$, contradicting step 1.1.

If $p\in \mathrm{\partial }B\setminus \mathrm{\partial }A$, reverse the time orientation to $-N$. This exchanges $A$ with $B$, preserves $E$, causal convexity and the Cauchy condition, and the same geodesic and one-sided-extension suppliers apply. Steps 2.1 and 3.1 then give the time-reversed contradiction. Step 1.2 places every boundary point on at least one of the two boundaries, and both single-boundary cases have now been excluded. Therefore ${\mathrm{\partial }}_{\mathrm{\Sigma }}C\subset \mathrm{\partial }A\cap \mathrm{\partial }B$.