Bounded Observable Universe and General Relativity

Question

The book General Relativity for Mathematicians by Sachs and Wu defines a "spacetime" as a connected 4-dimensional, oriented and time oriented Lorentzian manifold. I emphasize the word "connected." There are those that say the observable universe is bounded, because the universe is expanding faster than the speed of light. A bounded observable universe that is isolated from the rest of the universe, strikes me as a disconnected manifold, and would not seem to comport with general relativity's notion of spacetime. Is this correct?

Following up on the above point, wouldn't there have to be some sort of discontinuity, at the moment when the universe splits off into isolated pieces? How many discontinuities are being hypothesized? Does general relativity contemplate discontinuities in spacetime and gravity?

Answer 1

The universe may be bounded (or it might not be) and parts of it are receding faster than the speed of light (in some sense) but these two notions are independent. It would be possible to have a universe that unbounded and expanding, or bounded and stationary etc.

Similarly it seems that "connected" means connected in the topological sense: the only simultaneously open and closed non-empty subspace of the universe is the whole universe. This is independent of any physical laws such as "information can't travel faster than the speed of light" which may make parts of the manifold inaccessible to us.

The universe could be said to be the disjoint union of the observable and non-observable universe, but these sets are not both open in the topological sense.