Does space shrink or is it just that all matter is combined into a single point? Is the single point “everywhere” like when discussing the big bang or is it actually at a single place?
If the question doesn’t make sense, please feel free to say so. It won’t hurt my feelings.
Space would shrink, just like with the Big Bang, but in reverse, leading to everything getting closer and closer together. You can’t quite say that they all reach a single point, because there’s a singularity there which isn’t formally considered part of the spacetime (i.e., there’s no “there” there), but they would get arbitrarily close together as you approach that singularity.
Current evidence strongly suggests that this won’t be the fate of the Universe as a whole, but the question is still relevant, because what would happen in a Big Crunch universe is exactly the same as what happens (so far as our mathematical models can determine) inside of a black hole.
My understanding is that space only exists to accommodate energy/matter - thus, when all energy/matter converges toward a single point, space seems to shrink as well.
No, you can have space without energy/matter. In fact, some of the standard cosmological models have space but no matter or energy (obviously, these are toy models, not meant to represent the actual Universe).
You might look into heat death of the universe or IIRC the Big Rip as alternate endings
I want to buy one of those toys!!! 
Hm. I am not here to argue pointlessly, but I like things to be clear where clarity is of course a function of my level of understanding.
When you say that I can have space without energy/matter, do you mean I (or we as a whole, i.e. mankind) can observe such space without energy/matter somewhere in the observable universe?
If you refer to mere theoretical models, that is constructions that can conceptualized but not observed as physical phenomena, in what respect are these speculations superior to the ontological argument?
But you feel that space only exists to accommodate matter/energy. That gets perilously close to a existence by necessity argument itself.
The point about a universe without energy/matter is that we can build the mathematical definition and derive properties for it without the need to populate it with matter. This suggests that there is no clear derivation for why space has a necessary existence tied to energy/matter.
We don’t live in such a universe. And the nature of space is clearly filled with fields that are tied to energy and matter (stress-energy tensor being the manifestation of this on spacetime itself.) But if we remove these fields from our model, spacetime doesn’t vanish. Which certainly suggests there is no necessary existence.
What does a scientist ‘feel’? Does a scientist feel space is distinct from body? I thought it was an abandoned Newtonian concept and modern science no longer regards space as a mere frame or shell of matter/energy.
To avoid going around in circles - you wrote: My understanding is that space only exists to accommodate energy/matter.
That is where what I wrote comes from. So ignore my writing “feel” replace it with “understand” as you first wrote. I’m perhaps parsing your sentence incorrectly, but you seemed to be suggesting an argument from necessity. Chronos pointed out that there was no need, space can be described that does not frame anything else.
It is perhaps worth noting that modern physics doesn’t talk about space anyway, space-time is the concept. An empty space-time has some properties, but is a bit boring. Reasoning about the nature of empty space goes back to Mach. Unless you have something inside your space time, you can’t reason much. There is no movement for a start. That gets you quickly to Mach’s principle, and perhaps that is why you might think that spacetime is there for matter, but you could equally say that matter is there to make sense of spacetime.
From a handwaving point of view, we know we don’t know how to unify QM with relativity, and we have thus no unification of the nature of space-time with the fields within it - although we do know that those fields bend spacetime - so there is a tantalising promise of something big in there. You can come up with no end of fanciful ideas about what might be.
It may just be my own lack of understanding, but I find comparisons with the Big Bang or Big Crunch singularity and black hole singularities to be troublesome. A black hole singularity distinctly exists in spacetime, and exerts profound effects on it. It creates an event horizon – which could be thought of as a transition point between space and time – whose size depends on the mass of the singularity. Whereas the BB/BC singularity does not exist in spacetime and surely cannot be described as having mass; as I understand it, the Higgs field did not even exist until about 10[sup]-32[/sup] seconds after the Big Bang. So ISTM that even the most trivial description of a singularity – a mathematical point of infinite density – doesn’t apply here, because “point” implies space, and “density” implies mass.
The difference is that theoretical models supply the possibility of an ‘empty’ universe, but nobody’s arguing that this possibility entails their necessity.
The Higgs field is in no way necessary for the existence of mass, and the vast majority of mass that we’re familiar with is completely unrelated to the Higgs field.
If the universe is a Big Crunch one, then we’re all already within the event horizon.
This question may no longer be moot.
In 1998, two independent studies were released that indicate **the expansion of the universe is actually ACCELERATING and not slowing down ** as one would expect.
Wikipedia: "The expansion of the universe is thought to have been accelerating since the universe entered its dark-energy-dominated era roughly 5 billion years ago.[9][notes 1] Within the framework of general relativity, an accelerating expansion can be accounted for by a positive value of the cosmological constant Λ, equivalent to the presence of a positive vacuum energy, dubbed “dark energy”.
The main difference between a big bang singularity, a big crunch singularity and a black hole singularity, is that the big bang singularity is the past-inextensibility of all timelike curves, the big crunch singularity is the future-inextensibility of all timelike curves and a black hole singularity is the future-inextensibility of some timelike curves. Though if there is a black hole in a universe with a big crunch you can’t really say there is a separate black hole singularity.
I should say “causal curve” (timelike and light curves) rather than “timelike curve” and by “inextensibility” I mean that curves are inextendible beyond some finite value of the affine parameter.
I may nurture the wrong idea about knowledge but I don’t think factual assertions should ground on epistemic modality, where both possibility and necessity stem from assumptions.
Steven Hawking once asked “What lies North of the North Pole?”
A non-rotating non-charged black hole does not have a point mass at the center. It does not have really have center at all as that point singularity is not part of spacetime but as undefined. Under Schwarzschild geometry the point singularity can be thought of as the limit and the falling particle geodesic path as an asymptote.
Oppenheimer provides a simplified collapse model that would describe a slightly more realistic but still simple black hole.
The oppenheimer/snyder metric can be used up to the point the singularity forms, and it is easier to consider the mass of what ever collapsed as the same mass as the resulting black hole. The event horizon can/will exist before the singularity forms under OS.
A black hole singularity is probably better thought of as not as a point in spacetime but undefined in spacetime. ADM Mass is probably what you are looking for after the singularity forms.
BB/BC/BH singularities can be unified to help with conceptualization by thinking of the singularity as being* geodesically incomplete in finite time*.
In the case of the Big Bang time-like geodesics just don’t have a past and in the case of BC they have no future. Those geodesics just end over finite periods of time.
In the case of a Black Hole you cannot describe a falling particle’s path when it reaches the singularity. As falling particles path over finite time is also geodesically incomplete.
It’s not a factual assertion, but a counterfactual one (nobody is saying our universe is devoid of matter, but if it were…), and there’s nothing but modality to ground them on. According to our best theories, such a universe is possible (a possible world with these properties exists). If you forbid such claims, you forbid any counterfactuals based on current theory. And then, we may as well pack up with this whole science stuff (since we couldn’t make any claim of the form ‘if [experiment], then [outcome]’).
I just want to say thanks for your comments and for that link. I haven’t read the paper yet but I will. My first reaction was surprise that black holes had been theorized that long ago (1939), but indeed they were. I see that Subrahmanyan Chandrasekhar published his calculations on the famous Chandrasekhar limit in 1930. For some reason I had thought it was much later.