Editing Nerve complex (section)

==Basic definition==
Let <math>I</math> be a set of indices and <math>C</math> be a family of sets <math>(U_i)_{i\in I}</math>. The '''nerve''' of <math>C</math> is a set of finite subsets of the index set ''<math>I</math>''. It contains all finite subsets <math>J\subseteq I</math> such that the intersection of the <math>U_i</math> whose subindices are in <math>J</math> is non-empty:''<ref name=":0">{{Cite Matousek 2007}}, Section 4.3</ref>{{Rp|page=81|location=}}''
:<math>N(C) :=  \bigg\{J\subseteq I: \bigcap_{j\in J}U_j \neq \varnothing, J \text{ finite set} \bigg\}.</math>
In Alexandrov's original definition, the sets <math>(U_i)_{i\in I}</math> are [[Open set|open subsets]] of some topological space <math>X</math>.

The set <math>N(C)</math> may contain singletons (elements <math>i \in I</math> such that <math>U_i</math> is non-empty), pairs (pairs of elements <math>i,j \in I</math> such that <math>U_i \cap U_j \neq \emptyset</math>), triplets, and so on. If <math>J \in N(C)</math>, then any subset of <math>J</math> is also in <math>N(C)</math>, making <math>N(C)</math> an [[abstract simplicial complex]]. Hence N(C) is often called the '''nerve complex''' of <math>C</math>.