Light takes a finite time to reach us so we see objects as they were at some instant in the past. Our view of objects in a room is about 6×10-9 second old, of the Moon it is about one second old; of the Sun it is some eight minutes old, and of Sirius it is about nine years old. Events that seem to be happening simultaneously from one place must therefore seem to occur at different times when seen from a different place. Time must therefore be measured in a relative manner and this is achieved through the concept of space-time.
Euclid (c.325-265 BC) postulated that two parallel lines when extended indefinitely will never meet, so Euclidean geometry is flat. In space-time, however, the familiar laws of geometry no longer apply in regions where the gravitational fields are strong. In 1829 Nikolai Lobachevsky proposed a new geometry in which parallel lines when extended were allowed to diverge. His geometry has properties similar to the surface of a saddle-shaped plane in which parallel lines diverge when extended, giving rise to infinite hyperbolic geometry. Both flat and hyperbolic geometries are therefore open geometries. In 1894 Georg Riemann devised a geometry in which parallel straight lines when extended were allowed to converge. His geometry has properties similar to the surface of a sphere. Thus a surface of a sphere has spherical or closed geometry.
In the simplest cosmological models, space-time curvature is based on the mean density of matter, i.e. the density obtained if all the matter concentrated in the galaxies was spread evenly across the Universe. If the mean density of a universe is high enough then gravitational attraction will pull this universe inward and give it a positive curvature, meaning that its space-time will curve in on itself: it will have a finite size and a finite lifetime and will be a closed universe. If the mean density is less than a critical value then the universe will have a negative curvature, meaning that it will be infinite and expand forever: it will be an open universe. It is not suggested that an open universe would look like a saddle, anymore than a closed universe would look like a sphere. A universe which has critical density is spatially flat and infinite in both space and time is an open universe.
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