Compact-target null generators and the required endpoint conditions

Apply [F3]–[F4] using the globally hyperbolic spacetime $E$, compact smooth target $C$, and the strict temporal separation in the Statement. These supply a nonconstant future null geodesic to a point of $C$. Apply [F5] to get containment of its whole closed segment in ${\mathrm{\partial }}_E{I}_E^{-}(C)$; [F6] gives endpoint orthogonality and [F7] excludes interior focal points measured from that endpoint. Each use retains precisely the same domain and causal relations.

Put $P:=I_E^{-}(C)={\bigcup }_{c\in C}{I}_E^{-}(c)$. Each set in this union is open by [F9], so $P$ is open by [F11]. If $x\in I_E^{-}(P)$, some $y\in P$ and $c\in C$ satisfy $x{\ll }_{E}y{\ll }_{E}c$. Positively affinely reparametrize the two joining timelike curves to adjoining compact intervals and concatenate them using [F10]. This gives $x{\ll }_{E}c$, hence $I_E^{-}(P)\subseteq P$.

Conversely, if $x\in P$, choose a future timelike curve $\gamma :[a,b]\to E$ from $x$ to some $c\in C$ and an interior parameter $s\in (a,b)$. Restriction to $[a,s]$ and $[s,b]$ preserves local absolute continuity and the nonzero future timelike velocity condition almost everywhere, directly from the curve definition in [F1]; thus $x{\ll }_E\gamma (s){\ll }_{E}c$. It follows that $\gamma (s)\in P$ and $x\in I_E^{-}(P)$. Together with step 1.2 this gives $I_E^{-}(P)=P$. Therefore $P$ is an open past set in [F8]'s sense; reversing the time orientation makes it an open future set, with the same topological boundary. Apply [F8] to obtain the local Lipschitz property of ${\mathrm{\partial }}_{E}P$. The argument does not choose a null-normal branch or assert regularity of an endpoint map; those remain the separate obligations in the Statement.