Nerve complex

Constructing the nerve of an open good cover containing 3 sets in the plane.

In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov[1] and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings. It captures many of the interesting topological properties in an algorithmic or combinatorial way.[2]

  1. ^ Aleksandroff, P. S. (1928). "Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung". Mathematische Annalen. 98: 617–635. doi:10.1007/BF01451612. S2CID 119590045.
  2. ^ Eilenberg, Samuel; Steenrod, Norman (1952-12-31). Foundations of Algebraic Topology. Princeton: Princeton University Press. doi:10.1515/9781400877492. ISBN 978-1-4008-7749-2.