Why is “closed\ necessarily spherical, critical is flat, etc?
“Closed” referring to a spherical universe is a purely geometric association: a spherical geometry is called closed, because you can go in a straight line and end up on a closed path on its surface; flat is called flat for obvious reasons, and a hyperbolic geometry is called open because your straight line will go off to infinity. The association of these geometries with the fate of the universe is actually a somewhat different issue. The fate of the universe depends on mass density of the universe (under the assumption that only gravity is acting- when you throw dark energy in the situation is different!). The overall equations describing the universe’s geometry show that the geometry is closed when the mass density exceeds the critical density, flat when the mass density is exactly equal to the critical density, and open when the mass density is less than the critical density. The math here is beyond the scope of this class, but I hope it’s plausible: “closed”, “flat” and “ope
