There is good evidence, for example from the APM survey, and the COBE satellite, that the universe looks the same in all directions. Distant galaxies for example appear to be distributed smoothly over the sky. This property is called isotropy.
Homogeneity:
The evidence for the universe looking the same at all positions is less compelling. To find how galaxies are distributed about another point in space we need to measure the positions of many galaxies in three dimensions. This is the job of redshift surveys. Using the direction in which a galaxy is observed and its redshift as a measure of its distance, the position of the galaxy can be determined. Currently each survey sees structures that are as large as the region that is observed. Thus there is no evidence for homogeneity of the light distribution. As we dont know exactly how the distribution of light coming from and object corresponds to the distribution of mass within it, we are unable to comment on the homogeneity of the mass distribution. We will assume that the universe is homogeneous, but we should remember that we have made this assumption. If our model goes horribly wrong, or amazingly right, we have to remember the didgy bits of the argument.
Robertson-Walker Metric:
Taking the universe to be isotropic and homogeneous fixes the metric to be the Robertson-Walker metric. Metric and cosmological redshift.
