Gluing two developments along their maximal common development: the quotient is a development containing both

(Equivalence relation.) Reflexivity and symmetry of $\sim$ are built into its definition; for transitivity let $z\sim w$ and $w\sim v$. If $z=w$ or $w=v$ then $z\sim v$ directly. Otherwise, up to exchanging the roles of the two members, $z={\iota }_M(p)$, $w={\iota }_{M^{\mathrm{\prime }}}(\phi (p))$, $p\in U$; and $w\sim v$ with $w\in M^{\mathrm{\prime }}$ forces $v=w$ or $v={\iota }_M({\phi }^{-1}(w))={\iota }_M(p)$; in both cases $z\sim v$.

($p_M$ and $p_{M^{\mathrm{\prime }}}$ are open injections.) For open $A\subseteq M$ the saturation is ${\pi }^{-1}(p_M(A))={\iota }_M(A)\cup {\iota }_{M^{\mathrm{\prime }}}(\phi (A\cap U)),$ which is open in $Z={\iota }_M(M)\bigsqcup {\iota }_{M^{\mathrm{\prime }}}(M^{\mathrm{\prime }})$ because $A$ and $A\cap U$ are open and $\phi$ is a homeomorphism onto its image ([L2]); hence $p_M(A)$ is open in $Q$ ([L5]), and similarly for $p_{M^{\mathrm{\prime }}}$. If $p_M(p)=p_M(q)$ with $p\mathrm{\ne }q$, then ${\iota }_M(p)\sim {\iota }_M(q)$ requires ${\iota }_M(p)={\iota }_{M^{\mathrm{\prime }}}(\phi (p))$ or ${\iota }_M(q)={\iota }_{M^{\mathrm{\prime }}}(\phi (q))$, impossible inside the disjoint union, or ${\iota }_M(p)={\iota }_M(q)$, i.e. $p=q$; so $p_M$ is injective, and similarly $p_{M^{\mathrm{\prime }}}$. Both are continuous as compositions of the continuous inclusions with $\pi$ ([L5]).

(The overlap.) A class has representatives in both members exactly when it has a source representative in $U$, so $p_M(U)=p_{M^{\mathrm{\prime }}}(\phi (U))=p_M(M)\cap p_{M^{\mathrm{\prime }}}(M^{\mathrm{\prime }})$. Consequently $Q=p_M(M\setminus U)\bigsqcup p_M(U)\bigsqcup p_{M^{\mathrm{\prime }}}(M^{\mathrm{\prime }}\setminus \phi (U))$ as a disjoint set decomposition. Only the two full member images and their intersection are asserted open; the complements may contain boundary points and need not be open.

(Smooth atlas.) Let $\{(V_i,{\varphi }_i){\}}_{i\in \mathbb{N}}$ and $\{(V_k^{\mathrm{\prime }},{\varphi }_k^{\mathrm{\prime }}){\}}_{k\in \mathbb{N}}$ be countable atlases of $M$ and $M^{\mathrm{\prime }}$ ([L2]); their existence uses ${\mathrm{A}\mathrm{C}}_{\omega }$ applied to the countable bases of the two manifolds. Then $\{(p_M(V_i),{\varphi }_i\circ p_M^{-1}){\}}_{i\in \mathbb{N}}\cup \{(p_{M^{\mathrm{\prime }}}(V_k^{\mathrm{\prime }}),{\varphi }_k^{\mathrm{\prime }}\circ p_{M^{\mathrm{\prime }}}^{-1}){\}}_{k\in \mathbb{N}}$ is a countable open cover of $Q$ (step 3.1) by homeomorphisms onto open subsets of ${\mathbb{R}}^4$. Its transition maps are of the form ${\varphi }_{i_0}^{-1}\circ {\varphi }_{i_1}$, the primed analogue, or ${\varphi }_k^{\mathrm{\prime }}\circ \phi \circ {\varphi }_i^{-1}$ (suitably restricted) on the overlap in $p_M(U)$; the first two are smooth because the atlases are smooth, and the third is smooth because $\phi :(U,g{\mathrm{\mid }}_U,\psi {\mathrm{\mid }}_U)\to M^{\mathrm{\prime }}$ is a diffeomorphism onto the open set $\phi (U)$ ([L2]). Hence $Q$ is a smooth four-manifold with this structure, and it is the unique structure making $p_M,p_{M^{\mathrm{\prime }}}$ diffeomorphisms onto their images, since any such structure has these charts in its atlas.

(Hausdorffness.) Two distinct classes represented in the same member are separated by images of disjoint open neighborhoods there, using step 2.1. For the remaining case choose representatives $z\in M\setminus U$ and $w\in M^{\mathrm{\prime }}\setminus \phi (U)$. If $z\mathrm{\notin }\bar{U}$, choose an open neighborhood $V\ni z$ disjoint from $U$. Its image has no representative in $M^{\mathrm{\prime }}$, so is disjoint from the open full image $p_{M^{\mathrm{\prime }}}(M^{\mathrm{\prime }})$ containing $[w]$. The symmetric argument applies when $w\mathrm{\notin }\overline{\phi (U)}$. Otherwise $z\in \mathrm{\partial }U$ and $w\in \mathrm{\partial }\phi (U)$. If every pair of their image neighborhoods met, then for every open $V\ni z$, $V^{\mathrm{\prime }}\ni w$ there would be $x\in V\cap U$ with $\phi (x)\in V^{\mathrm{\prime }}$, by the exact overlap relation. This would be a corresponding boundary pair, contrary to [L1]. Thus the classes have disjoint neighborhoods.

(Second countability.) A homeomorphism transports a countable basis to a countable basis, so transporting the countable bases of $M$ and $M^{\mathrm{\prime }}$ along $p_M$ and $p_{M^{\mathrm{\prime }}}$ gives two countable bases of the open sets $p_M(M)$ and $p_{M^{\mathrm{\prime }}}(M^{\mathrm{\prime }})$; their union is at most countable and is a basis of $Q$ because every open subset of $Q$ is a union of its traces in the two open parts, which are unions of members of these bases. Hence $Q$ is second countable (Second countability: an at most countable basis for the topology, Finite, countably infinite, countable, uncountable).

(Metric, full matter and a future orientation field.) The pushed metrics and each fixed-model matter tensor agree on the overlap because $p_{M^{\mathrm{\prime }}}\circ \phi =p_M$ and $\phi$ preserves them. Thus they define smooth $\bar{g},\bar{\psi }$ on $Q$. Future cones also agree, since the common map preserves the time orientation as a cone. The representing fields $N,N^{\mathrm{\prime }}$ themselves need not agree there. By steps 4.2-4.3, $Q$ is a Hausdorff second countable smooth manifold. Under Countable Choice, a subordinate smooth partition $({\rho }_M,{\rho }_{M^{\mathrm{\prime }}})$ for its two-member open cover exists by Smooth partitions of unity exist on manifolds. Define $\bar{N}={\rho }_M(p_M{)}_{\ast }N+{\rho }_{M^{\mathrm{\prime }}}(p_{M^{\mathrm{\prime }}}{)}_{\ast }{N}^{\mathrm{\prime }}$, extending each supported term by zero. This is smooth, and at each point is a nonzero future timelike vector because the weights sum to one and the timelike cone is convex (The timelike set has exactly two convex components, described in orthonormal coordinates). Both member maps therefore preserve the future cones of $\bar{N}$. All field equations depend on the metric and full fixed-model matter, rather than the arbitrary orientation representative.

(The field equations and the induced data.) The equations of the data (Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a)4) are local and hold on $M$ and $M^{\mathrm{\prime }}$, so they hold on $Q$ because $Q$ is covered by the images of the two isometries (step 5.1). The map $\bar{j}=\pi \circ {\iota }_M\circ j=p_M\circ j:\mathrm{\Sigma }\to Q$ is an embedding with spacelike image, since $p_M$ is a diffeomorphism onto an open set and $j$ has these properties in $M$; its induced metric, second fundamental form and matter data are $(h,K,{\psi }_0)$ because $p_M$ is a data-fixing isometry ([L2]).

(The Cauchy property.) An inextendible causal curve in $Q$ has a maximal portion in any open member image that it meets. By [L4] that portion is inextendible in the member and hence meets its Cauchy surface, so the whole curve meets $\bar{j}(\mathrm{\Sigma })$ at least once. For uniqueness, the sets $Q^{\pm }=p_M(I_M^{\pm }(\mathrm{\Sigma }))\cup p_{M^{\mathrm{\prime }}}(I_{M^{\mathrm{\prime }}}^{\pm }({\mathrm{\Sigma }}^{\mathrm{\prime }}))$ are open and disjoint, and $Q=Q^{-}\bigsqcup \bar{j}(\mathrm{\Sigma })\bigsqcup Q^{+}$: this follows from the member side decompositions and their agreement on the common subdevelopment by causal convexity and future cone preservation. The surface is closed since its trace in each open member image is the closed member Cauchy surface. Near any surface point a future causal curve crosses from the past side to the future side, and has at most one local parameter visit, by a spacelike slice chart and strict temporal-coordinate increase. Thus the set of surface visits in a compact parameter interval is closed and locally isolated, hence finite. If two visits existed choose consecutive ones $t_1<t_2$. The connected intervening image, which avoids the surface, must lie in one of $Q^{-},Q^{+}$. Near $t_1$ it lies in $Q^{+}$, whereas near $t_2$ it lies in $Q^{-}$. This is impossible. Therefore there is exactly one visit.

(Conclusion.) The preceding steps exhibit $(Q,\bar{g},\bar{N},\bar{j},\bar{\psi })$ as a time-oriented second countable smooth Lorentzian four-manifold satisfying the field equations on $Q$, with $\bar{j}(\mathrm{\Sigma })$ a spacelike embedded hypersurface meeting every inextendible causal curve exactly once and carrying the induced data of the same initial data: a development of the data. The maps $p_M:M\to Q$ and $p_{M^{\mathrm{\prime }}}:M^{\mathrm{\prime }}\to Q$ are diffeomorphisms onto open subsets, isometric, time-orientation preserving, matter preserving and satisfy $p_M\circ j=\bar{j}=p_{M^{\mathrm{\prime }}}\circ j^{\mathrm{\prime }}$ (step 3.1), so they are data-fixing development embeddings. This proves (a), (b) and (c).