Compactness of closed advanced-retarded support overlaps under Countable Choice

(Elementary timelike coordinate neighborhoods.) At $x$ choose linear coordinates about $x$ in which ${\mathrm{\partial }}_0$ is future timelike at $x$. Shrinking the coordinate box, continuity of $g$ and $N$ gives $ϵ>0$ such that every vector with positive zeroth component and spatial-to-time coordinate ratio below $ϵ$ is future timelike throughout that box. Choose a point $p$ on the negative time axis and a smaller open box $O_x$ about $x$, with closures inside the first box, so that every straight coordinate segment from $p$ to any point of $O_x$ has such a tangent. Then $O_x\subseteq J^{+}(p)$ by the definition in [F1]. The reversed construction gives an open box about $x$ contained in $J^{-}(q)$ for a point $q$ on its positive time axis. This uses local smooth metric continuity and straight smooth segments, without a limit-curve, global chronology-openness or dependent-choice theorem.

(Finite compact enclosure.) Take the family of all pairs of boxes and past points furnished by step 1.1 over $K$. These boxes form an open cover of $K$; taking a finite subcover names finitely many $p_i$ with $K\subseteq {\bigcup }_i{J}^{+}(p_i)$. No past-point choice for an infinite family is made. Similarly take finitely many $q_j$ with $L\subseteq {\bigcup }_j{J}^{-}(q_j)$. Concatenating the finite causal segments as in [F1] proves transitivity directly, so $J^{+}(K)\cap J^{-}(L)\subseteq {\bigcup }_{i,j}(J^{+}(p_i)\cap J^{-}(q_j))\mathrm{.}$ The right side is a compact set $D$ by [F1] and [F2]. If $C\subseteq J^{+}(K)\cap J^{-}(L)$ is closed in $M$, then $C=C\cap D$ is closed in the subspace $D$, and hence compact by [F2]. In particular $A\cap B$ is closed in $M$ and has the required containment. If either input compact is empty the conclusion is immediate from the empty cover.