its simplices. It is usually denoted by or . Support The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted . Closure, star, and link Let K be a simplicial complex and let S be a collection of simplices in K. The closure of S (denoted ) is the smallest simplicial subcomplex of K that contains each simplex in S. is obtained by repeatedly adding to S each face of every simplex in S. The star of S (denoted ) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. The star of S is generally not a simplicial complex itself, so some authors define the closed star of S (denoted ) as the closure of the star of S. The link of S (denoted ) equals . It is the closed star of S minus the stars of all faces of S. Algebraic topology In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see , , ). Combinatorics Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence , where fi is the number of (i−1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6,