I've been studying singularities in GR, and (obviously), came across PST.

Let us state it as the following:

Let $(M, g)$ be a connected globally hyperbolic spacetime with a noncompact Cauchy hypersurface $S$, satisfying the null energy condition. If $S$ contains a trapped surface $\Sigma$, then $(M,g)$ is singular.

I'll suppose the reader is familiar with the definitions used above. For the purpose of my question, I do believe intuition on those is probably sufficient. If my assumptions turn out to be inappropriate, I'll be happy to edit the question.

Anyway, I'm having a hard time coming around as to why $S$ needs to not be compact. Simple as that.

What does it represent, for the theorem as a whole?

Any help will be greatly appreciated. Thank you in advance.