Editing Abstract simplicial complex (section)

==Examples==
1. Let ''V'' be a finite set of [[cardinality]] {{math|''n'' + 1}}. The '''combinatorial ''n''-simplex''' with vertex-set ''V'' is an ASC whose faces are all nonempty subsets of ''V'' (i.e., it is the [[power set]] of ''V''). If {{math|''V'' {{=}} ''S'' {{=}} {0, 1, ..., ''n''},}}  then this ASC is called the '''standard combinatorial ''n''-simplex'''.

2. Let ''G'' be an undirected graph. The '''[[clique complex]]''' '''of ''G''''' is an ASC whose faces are all [[Clique (graph theory)|cliques]] (complete subgraphs) of ''G''. The '''independence complex of ''G''''' is an ASC whose faces are all [[Independent set (graph theory)|independent sets]] of ''G'' (it is the clique complex of the [[complement graph]] of G).  Clique complexes are the prototypical example of [[flag complex]]es. A '''flag complex''' is a complex ''K''  with the property that every set, all of whose 2-element subsets are faces of ''K'', is itself a face of ''K''.

3. Let ''H'' be a [[hypergraph]]. A [[Matching in hypergraphs|matching]] in ''H'' is a set of edges of ''H'', in which every two edges are [[Disjoint sets|disjoint]]. The '''matching complex of ''H''''' is an ASC whose faces are all [[Matching in hypergraphs|matchings]] in ''H''. It is the [[independence complex]] of the [[Line graph of a hypergraph|line graph]] of ''H''.

4. Let ''P'' be a [[partially ordered set]] (poset). The '''order complex''' of ''P'' is an ASC whose faces are all finite [[Total order#Chains|chains]] in ''P''. Its [[Homology (mathematics)|homology]] groups and other [[Topological property|topological invariants]] contain important information about the poset ''P''.

5. Let ''M'' be a [[metric space]] and ''δ'' a real number. The '''[[Vietoris–Rips complex]]''' is an ASC whose faces are the finite subsets of ''M'' with diameter at most ''δ''. It has applications in [[homology theory]], [[hyperbolic group]]s, [[image processing]], and [[mobile ad hoc network]]ing. It is another example of a flag complex.

6. Let <math>I</math> be a square-free [[monomial ideal]] in a [[polynomial ring]] <math>S = K[x_1, \dots, x_n]</math> (that is, an ideal generated by products of subsets of variables). Then the exponent vectors of those square-free monomials of <math>S</math> that are not in <math>I</math> determine an abstract simplicial complex via the map <math>\mathbf{a}\in \{0,1\}^n \mapsto \{i \in [n] : a_i = 1\}</math>. In fact, there is a bijection between (non-empty) abstract simplicial complexes on {{math| ''n''}} vertices and square-free monomial ideals in {{math| ''S''}}. If <math>I_{\Delta}</math> is the square-free ideal corresponding to the simplicial complex <math>\Delta</math> then the [[Quotient ring|quotient]] <math>S/I_{\Delta}</math> is known as the [[Stanley–Reisner ring]] of <math>{\Delta}</math>.

7. For any [[open covering]] ''C'' of a topological space, the '''[[nerve complex]]''' of ''C'' is an abstract simplicial complex containing the sub-families of ''C'' with a non-empty [[intersection]].