Nerve of a covering
- 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, if and only if
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 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.
- Paul Alexandroff Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.
- Samuel Eilenberg and Norman Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952, p. 234.