In topology, the nerve of an open covering is a construction of an abstract simplicial complex from an open covering of a topological space X.
The notion of nerve was introduced by Pavel Alexandrov.
Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:
Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.
In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-sphere with two contractible sets U and V, in such a way that N is an abstract 1-simplex. However, if we also insist that the open sets corresponding to every intersection indexed by a set in N is also contractible, the situation changes. This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of N.