Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.

**Definition 1:** A triplet $ (\Sigma,h,k) $ is called an **initial data set** if $ (\Sigma,h) $ is a Riemannian $ 3 $-manifold, $ k $ is a symmetric $ 2 $-form on $ \Sigma $, and both $ h $ and $ k $ satisfy the so-called Einstein constraint equations for vacuum (we do not need to know what these equations are). $ \quad \spadesuit $

**Definition 2:** Let $ (\Sigma,h,k) $ be an initial data set. A triplet $ (M,g,i) $ is called a **vacuum development of $ (\Sigma,h,k) $** if

$ (M,g) $ is a spacetime manifold (i.e., a Lorentzian $ 4 $-manifold) that satisfies the vacuum Einstein Field Equations;

$ i $ is a smooth embedding of $ \Sigma $ into $ M $ such that $ {i^{*}}(g) = h $ and $ {i^{*}}(\Pi) = k $, where $ \Pi $ is the second fundamental form (i.e., the extrinsic curvature) of $ \Sigma $ in $ M $; and

$ i[\Sigma] $ is a smooth Cauchy surface in $ M $. $ \quad \spadesuit $

The Cauchy Problem in General Relativity is to prove the existence of a vacuum development for every initial data set. The proof proceeds by working locally on small patches of the initial data set and ends by performing some gluing to obtain a global solution.

It is a well-known result from the theory of hyperbolic PDE’s that if $ U $ is a local coordinate patch in $ \Sigma $ (i.e., $ U $ is homeomorphic to an open subset of $ \mathbb{R}^{3} $), then we can find a vacuum development $ (M_{U},g_{U},i_{U}) $ of the initial data set $ (U,h|_{U},k|_{U}) $ such that $ M_{U} $ is an open neighborhood of $ U \times \lbrace 0 \rbrace $ in $ U \times \mathbb{R} $ and $ {i_{U}}[U] = U \times \lbrace 0 \rbrace $.

Knowing now that local vacuum developments exist, we have to glue them together in order to obtain a global-in-space vacuum development of $ (\Sigma,h,k) $. Once the Hausdorff property is established, paracompactness is for free by a 1968 result of Robert Geroch, which states: If a smooth $ 4 $-manifold is Hausdorff and admits a smooth Lorentz metric, then it is automatically paracompact.

In Robert Wald’s book *General Relativity*, the following method for gluing is proposed:

By the paracompactness of $ \Sigma $ as a $ 3 $-manifold, we can find a locally finite open cover $ \mathcal{U} $ of $ \Sigma $ consisting of coordinate patches. As mentioned above, for each $ U \in \mathcal{U} $, we have a vacuum development $ (M_{U},g_{U},i_{U}) $ of $ (U,h|_{U},k|_{U}) $.

For each $ p \in \Sigma $, define $ \mathcal{U}_{p} := \lbrace U \in \mathcal{U} ~|~ p \in U \rbrace $. We have from (1) that $ \mathcal{U}_{p} $ is a finite collection for each $ p \in \Sigma $.

For each $ p \in \Sigma $, let $ W_{p} \subseteq \Sigma $ be a neighborhood of $ p $ such that $ W_{p} $ is contained in all the open sets belonging to $ \mathcal{U}_{p} $, i.e., $ W_{p} \subseteq \bigcap \mathcal{U}_{p} $. Using a result known as ‘local geometric uniqueness’, we can then find a vacuum development $ (M_{p},g_{p},i_{p}) $ of $ (W_{p},h|_{W_{p}},k|_{W_{p}}) $ such that there exists an isometric embedding $ j_{p,U}: (M_{p},g_{p}) \to (M_{U},g_{U}) $ for each $ U \in \mathcal{U}_{p} $.

Using the embeddings $ j_{p,U} $ to make identifications, we can glue the $ (M_{p},g_{p}) $’s together in order to obtain a vacuum development of $ (\Sigma,h,k) $. Although Wald does not clearly describe this step, this is how I interpret it. For each $ U \in \mathcal{U} $, consider the union $ \tilde{M}_{U} := \displaystyle \bigcup_{p \in U} {j_{p,U}}[M_{p}] \subseteq M_{U} $. We wish to glue the $ (\tilde{M}_{U},g_{U}|_{\tilde{M}_{U}}) $’s together by identifying points as follows. Given $ U_{1},U_{2} \in \mathcal{U} $, $ x_{1} \in \tilde{M}_{U_{1}} $ and $ x_{2} \in \tilde{M}_{U_{2}} $, we say that $ x_{1} \sim x_{2} $ if and only if there exist a $ p \in U_{1} \cap U_{2} $ and a $ q \in M_{p} $ such that $ {j_{p,U_{1}}}(q) = x_{1} $ and $ {j_{p,U_{2}}}(q) = x_{2} $. Finally, define $ \displaystyle M := \bigsqcup_{U \in \mathcal{U}} \tilde{M}_{U} \bigg/ \sim $.

**Problem:** How do we show that $ M $ is Hausdorff?

This has eluded my best efforts for the past month, so I would appreciate it if anyone on MathOverflow could offer any insight. Thank you very much!

The Cauchy Problem in General Relativity(EMS, 2009). See proof of Theorem 16.6, p.179. $\endgroup$ – Igor Khavkine Jun 16 '13 at 16:03