This post in influenced by 3 new threads in our cosmology forum. Recent observational data favors positive curvature of our Universe. The question I have, however, is why positive curvature implies spatially finite Universe? Yes, it might look quite obvious if we embed curved space into higher dimensional flat space. But we can do it, but not must do it - we can work with GR without embedding, am I correct? Take Klein bottle as example. You can't correctly embed it into 3D space without having intersections with itself. Still it is a valid mathematical object, when you forget about intersections. I tend to believe that 'intersections' require additional axiom, saying that "when 2 different points of space can be mapped to the same point in the higher dimensional space, then it is the same point in lower dimensional space too". Without this axiom, space with positive curvature can be infinite - compare a circle and an infinite spring - they both have the same positive curvature... Also without embedding there is yet another option. For the space with constant positive curvature at least 2 finite configurations exist: sphere and half-sphere (sphere cut in half, with diametrically opposite points interconnected on a 'cut' side). So without embedding, full information about curvature doesn't give us the volume! So how do we know that our favorite 'balloon' is not cut in half?