Examples
- As an example, let V be a finite subset of S of cardinality n+1 and let K be the power set of V. Then K is called a combinatorial n-simplex with vertex set V. If V = S = {0, 1, 2, ..., n}, K is called the standard combinatorial n-simplex.
- The clique complex of an undirected graph has a simplex for each clique (complete subgraph) of the given graph. Clique complexes form the prototypical example of flag complexes, complexes with the property that every set of elements that pairwise belong to simplexes of the complex is itself a simplex.
- In the theory of partially ordered sets ("posets"), the order complex of a poset is the set of all finite chains. Its homology groups and other topological invariants contain important information about the poset.
- The Vietoris–Rips complex is defined from any metric space M and distance δ by forming a simplex for every finite subset of M with diameter at most δ. It has applications in homology theory, hyperbolic groups, image processing, and mobile ad-hoc networking. It is another example of a flag complex.
Read more about this topic: Abstract Simplicial Complex
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