Causal curves from a common chronological boundary obey two barriers

Fix $x\in M$. Apply [F3] with initial vector $v=N_x$. At $t=0$ both $g({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))$ and $g({\gamma }^{\mathrm{\prime }}(t),N_{\gamma (t)})$ equal $g_x(N_x,N_x)<0$. Smoothness and continuity make both negative for $0\le t\le ϵ$ for some $ϵ>0$. Thus for $0<t\le ϵ$ the smooth segment $\gamma {\mathrm{\mid }}_{[0,t]}$ is future timelike in the curve convention [F0], and $x\ll \gamma (t)$ while $\gamma (t)\to x$ as $t\downarrow 0$. The choice convention [A1] covers the inherited supplier hypotheses; this construction uses the specified vector $N_x$ and no additional choice of vectors.

If $p\le q$ and $q\in G$, some $e\in E$ satisfies $q\ll e$. Push-up gives $p\ll e$, contrary to $p\in \mathrm{\partial }G$ and openness of $G$. Thus $q\mathrm{\notin }G$. For any $q^{\mathrm{\prime }}\gg q$, push-up gives $p\ll q^{\mathrm{\prime }}$. Since $p\in \overline{F_{+}}$ and $I^{-}(q^{\mathrm{\prime }})$ is an open neighborhood of $p$, choose $f\in F_{+}\cap I^{-}(q^{\mathrm{\prime }})$. Its future-set property puts $q^{\mathrm{\prime }}\in F_{+}$. Apply step 1.1 at $q$: the endpoints $q^{\mathrm{\prime }}=\gamma (t)$ lie in $F_{+}$ and approach $q$, so $q\in \overline{F_{+}}$.

If $p\ll q$, the same open-endpoint argument puts $q\in F_{+}$: the open neighborhood $I^{-}(q)$ of $p\in \overline{F_{+}}$ meets $F_{+}$, and its future-set property applies. If $q\in \bar{G}$, openness of $I^{+}(p)$ would give $g\in G$ with $p\ll g$, and the past-set property would put $p\in G$, a contradiction. Hence $q\mathrm{\notin }\bar{G}$. Replacing $N$ by $-N$ reverses the relations and interchanges $G$ with $F_{+}$; the same arguments prove the past statements.