We do not appear to be in a closed universe. In a closed universe, the density is greater than the critical density. Such a universe will ultimately crash back down on itself (Big Crunch). Such a universe has a strange geometry. It has what is called positive curvature, a looks like a sphere. In a closed universe, when you get to very, very large scales, the circumference of a circle is smaller than 2*pi*R and all the angles in a triangle add up to MORE than 180.
A flat universe is one where the density of the universe is equal to the critical density.. In other words, it will stop expanding, but at an infinite distance. In a flat universe, geometry works the way you learned in school. The angles of a triangle add up to 180, and the circumference of a circle equals 2*pi*R no matter how you curl or manipulate a sheet of paper.
A universe which will expand for ever is called an open universe. In such a universe, the expansion energy dominates over the gravitational binding energy (Big Chill). The expansion continues to speed up and space is non-linear over large scales. An open universe has negatively curved space. It looks like a "saddle" or "potato chip". In such a universe, the circumference of a circle is GREATER than 2*pi*R and the angles in a right triangle add up to LESS than 180.
