I am trying to prove proposition 9.2.1 from text book "The large scale structure of spacetime" by Hawking and Ellis at page # 311, last line and got stuck in a topological/geometrical reasoning.

So we assume two things

a) spacetime is future asymptotically predictable from a partial cauchy surface$S$

b)$R_{ab}k^{a}k^{b} \geq 0$

So I am only writing the portion in which I am stuck

Let u be open neighborhood of M which is isometric to the neighborhood u' of$\partial M'$ in the conformal manifold$\tilde{M'}$ of an asymptotically empty and simple space$(M',g')$. Let$S'$ be a cauchy surface in M' which coincides with S on$u' \cap M'$. Then S'-u' is compact

My question is, under all this setting,How/Why S'-u' is compact ?

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