Nerve of an open cover

Definition
The nerve of an open cover of a topological space is defined as the following simplicial complex:


 * The vertices correspond to the members of the open cover (viz the open subsets in the cover)
 * A collection of vertices forms a simplex iff the intersection of the corresponding open sets is nonempty

As a functor
The nerve is actually a contravariant functor from the category of open covers of a topological space, to the category of simplicial complexes, with maps viewed upto homotopy. Namely, given two open covers such that one refines the other, we get a simplicial map between the associated simplicial complexes.