See my copyright notice for fair use practices. This final section will cover the fate of the universe. We observe that the universe is expanding and that gravity is slowing it down. The more mass there is, the more gravity there is to slow down the expansion.
Space curves back on itself and time has a beginning and an end.
If there is not enough matter, the universe will keep expanding forever. Space curves away from itself and time has no end. Instead of trying to add up all of the mass in the universe, a more reasonable thing to do is to find the density of a representative region of the universe. If the density is great enough, then the universe is closed. If the density is low enough, then the universe is open.