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==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''.
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