Compactness of spacetime: experiment and math

One mathematical comment which may be interesting:

The universal cover of simply connected boundaryless time-orientable four dimensional space-time manifold must be non-compact

The argument goes something like this: suppose we are given a compact closed manifold without boundary with 4 space-time dimensions that is simply connected. Its Euler characteristic is at least two (using Poincare duality). But if it is time-orientable it admits a non-vanishing vector field, which requires Euler characteristic 0. A contradiction.

The upside to this is that compactification procedures in GR end up with stuff that are even a bit worse than manifolds with boundaries, and are not closed 4 dimensional manifolds.

---

Note that there is also quite a big difference between a cosmological universe where the spatial Cauchy hypersurface is a closed compact manifold, and requiring that the entire space-time be compact. For starters, the assumption of a compact space-time would require necessarily that the universe ends after finite time, a rather pessimistic world-view to which I rather not subscribe.

For compactified Minkowski space see, e.g., Conformal Infinity.

This is sometimes useful to prove mathematical statements about general relativity and its relatives.

But to reach infinity or to get information from there takes infinitely long, given the finite speed of light. This is why we can never find out whether or not spacetime is compactified.