The selected exterior is connected and causally convex

For $x\in U$, take a smooth chart centered at $x$ and a future timelike vector at $x$. The coordinate straight segment in that direction is future timelike for all sufficiently small positive and negative parameter values, by continuity of the metric and the orientation field, and lies in $U$ because $U$ is open. It is an admissible timelike curve by [F2]. Its two halves give points $x_{-},x_{+}\in U$ with $x_{-}{\ll }_{M}x{\ll }_M{x}_{+}$, hence $x\in E$. Thus $U\subset E$. The sets $I_M^{\pm }(U)$ are unions of the open chronological futures or pasts of their points by [F1] and [F3], so $E$ is open as well.

If $U=\mathrm{\varnothing }$, then $E=\mathrm{\varnothing }$ and all conclusions hold, so consider nonempty $U$. Fix any $p\in E$. By [F1] there are $a,b\in U$ and future timelike curves from $a$ to $p$ and from $p$ to $b$. Reparametrize the first on $[0,1]$ using [F2]. Each interior point $z$ of that curve satisfies $a{\ll }_{M}z{\ll }_{M}p{\ll }_{M}b$ by restriction and concatenation, hence belongs to $E$; its endpoints also belong to $E$. Now take the set of all such timelike curves $\gamma :[0,1]\to M$ whose initial point lies in $U$ and whose final point lies in $E$. Their images lie in $E$ by the same argument, and every $p\in E$ is a final point of one. Each image is connected by the interval and continuous-image results in [F4], and meets the connected set $U$. The common-connected-core union result in [F4] therefore makes their union together with $U$, which equals $E$, connected. Taking all such curves selects no family of witnesses indexed by $E$.

Let a future causal curve have endpoints $p,q\in E$, and let $z$ be any point on it. Restriction gives $p{\le }_{M}z{\le }_{M}q$, including endpoint equalities. There are $a,b\in U$ with $a{\ll }_{M}p$ and $q{\ll }_{M}b$ by [F1]. The two push-up implications of [F3], under [A1], give $a{\ll }_{M}z{\ll }_{M}b$. Thus $z\in I_M^{+}(U)\cap I_M^{-}(U)=E$, proving causal convexity. The same conclusion for a past-directed curve follows by reversing its parameter and interchanging its endpoints.