of S, and give that direct limit the induced topology. Now give the subspace topology. Categorical definition Alternatively, let denote the category whose objects are the faces of and whose morphisms are inclusions. Next choose a total order on the vertex set of and define a functor F from to the category of topological spaces as follows. For any face X in K of dimension n, let be the standard n-simplex. The order on the vertex set then specifies a unique bijection between the elements of and vertices of , ordered in the usual way . If is a face of dimension , then this bijection specifies a unique m-dimensional face of . Define to be the unique affine linear embedding of as that distinguished face of , such that the map on vertices is order-preserving. We can then define the geometric realization as the colimit of the functor F. More specifically is the quotient space of the disjoint union by the equivalence relation that identifies a point with its image under the map , for every inclusion . Examples 1. Let V be a finite set of cardinality . 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 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 cliques (complete subgraphs) of G. The independence complex of G is an ASC whose faces are all independent sets of G (it is the clique complex of the complement graph of G). Clique complexes are the prototypical example of flag complexes. A flag complex is a complex K with the property that every set of elements that pairwise belong to faces of K is itself a face of K. 3. Let H be a hypergraph. A matching in H is a set of edges of H, in which every two edges are disjoint. The matching complex of H is an ASC whose faces are all matchings in H. It is the independence complex of the 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 chains in P. Its homology groups and other topological invariants contain important information about the