Given a valid classical GR spacetime manifold $M$ (i.e. 4D, Lorentzian, Hausdorff, paracompact, ?etc.), and $B\subset M$, a closed spatial subset (e.g. a closed ball at fixed $t$) to be excised,
[Aside: sorry - I try to be as precise as I can up front and this does seem to lead to unshort questions.]
The questions are:
a) Is the manifold $M' = M - B$ also a valid spacetime?
b) What is the argument/proof either way?
c) If $M'$ is not a valid spacetime, can it be repaired by some suitable surgery (other than putting back the excised subset)?
d) If $M$ is globally hyperbolic (GH), and $M'$ is either valid or can be repaired, is valid/repaired $M'$ also GH?
[NB: e.g. valid spacetime = Gödel; valid + GH = Minkowski]
My belief is that $M'$ is not a valid spacetime (or globally hyperbolic), but I can't demonstrate it*.
The objective of the question is to gain a better understanding of relevant technicalities, which I am aware of but do not comprehend; understanding the answers would be an entrée to better appreciation of theory.
Technical answers are welcome (esp. for citations and references to key theorems) even more so if they come with non-technical paraphrases that will help in understanding the technicalities.
One aspect of what I'm seeking is an insight into what you can and cannot do to a manifold in GR and why/why not. For example, in discussions of wormhole construction by surgery on Minkowksi/Schwarschild space I read the comment that after identification of surfaces (around excised open balls) "spacetime is geodesically complete" but whilst that is intuitively sensible, it is not obviously true to me.
* I believe that $M'$ is singular, having incomplete forward and backward directed timelike (and null ?+ spacelike) geodesics, but I'm also confused about completeness vs. extensibility and ? their relations to paracompactness, etc. I also don't think $M'$ is repairable because $B$ is closed and I haven't seen any reference in my searching to being able to rejoin open sets without adding "boundaries/closure". NB I am aware of extensive literature on how hard it is to define "singular" esp. w.r.t. geodesic completeness (e.g. Hawking, Penrose, Geroch, Earman, Curiel, etc.)... but I'd rather not go too deep.
[Optional subsidiary question. A potentially more complex instance might be: take two half-infinite world tubes starting/ending on $t=0$ (i.e. half-open $[0,+\infty)$ and $[0,-\infty)$) that have no spatial overlap and excise them from Minkowksi spacetime. Can the "ends" of the cylinders around $t=0$ be joined somehow to repair the spacetime?]