Nerve of a covering

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.^[1]

Given an index set I, and open sets U_i contained in X, the nerve N is the set of finite subsets of I defined as follows:

• a finite set J ⊆ I belongs to N if and only if the intersection of the U_i whose subindices are in J is non-empty. That is, if and only if

$\bigcap_{j\in J}U_j \neq \varnothing.$

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.

Notes

1. Paul Alexandroff Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.

References

• Samuel Eilenberg and Norman Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952, p. 234.