pcollins

An open universe does not mean that the universe is infinite, it simply means that the curvature for the observable universe is isomorphic to a saddle surface. We can't say with any certainty how mass-energy out of causal contact curves space-time, and if expansion continues unabated we never will. A spherical Earth was intuitive enough for anyone actually concerned with territory beyond the horizon. There's a reason why Earth's spherical isomorphism was realized (with a reasonable estimate of its circumference) by the early third century BC.

We start with 16 equations that relate the shape of space-time with matter or energy occupying. We can write this concisely as: [math]G_{ab} = \kappa T_{ab}[/math] (don't worry what [imath]\kappa[/imath] means, it's just a bunch of scaling constant anyway) [imath]G_{ab}[/imath] gives you the shape of space-time and [imath]T_{ab}[/imath] gives you the corresponding distribution of matter and energy. Let's say [imath]T_{ab}[/imath] is chosen to represent a planet of mass [imath]M[/imath]. Well, you compute [imath]G_{ab}[/imath] to get the associated curvature of space-time. You can observe space-time by firing a beam of light across the region [imath]G_{ab}[/imath] describes. The light will travel along the most direct path from its origin in the direction you aimed for. With the right instruments, you can convince yourself that the light bends as the surface it's traveling across does. Voi la, you've just observed space-time. Space-time isn't material. Material is encoded in the right side of the equation [imath]T_{ab}[/imath]. Space-time is hybrid construct of dimensions of measure. It's the thing that on which you place a coordinate system. Many physicists and cosmologists find it perfectly legitimate to talk of space-time as a physical thing, but personally I think that's confusing. They're not trying to say that space-time is an object with mass or energy, they're simply trying to say that space-time is coordinate-free constraint on the motion of objects described by [imath]T_{ab}[/imath].

Let's say the universe's topology is isomorphic (geometrically similar) to a sphere (it's likely not, but that's another topic). You don't want to think of yourself as an observer floating around in that sphere, but as an observer on the surface. Let's also pretend you're two dimensional; your natural perception of reality is constrained to the surface of that sphere. Now it should be obvious to you that this surface has only a finite area. It should also be obvious to you that if you pick a direction and start walking, you'll never run into a boundary--no edge to speak of. You'll just keep circumnavigating this sphere forever. These two facts about the surface of a sphere our 2D version of you has discovered are analogous to what cosmologists mean when they say the universe may be finite in one sense and infinite in another. The trick is to remember that we occupy a four dimensional space-time, a higher dimensional analogue to our sphere's surface, that might curve and twist in ways that permit its total content--a higher dimensional analogue to area and volume--to be finite while simultaneously permitting an observer to eternally move inertially (under no influence other than gravitation due to the curvature of space-time). Ghetto edit: replaced isotropic with isomorphic