Nerve of a cover

Summary

The nerve of a cover of a space is the simplicial complex that records which members of the cover meet. It has one vertex for each nonempty member, and a set of members spans a simplex when they have a common point. By the nerve theorem, a good cover has a nerve with the same homotopy type as the space. This turns topology into combinatorics, and nerves are also the basis of Čech cohomology.