Nerve of an open covering
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In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X. 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:
- the empty set belongs to N;
- a finite set J ⊆ I belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty. That is, iff
- <math>\bigcap_{j\in J}U_j \neq \emptyset.<math>
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 a 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 position 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.