in finetly many simplices, both topologies coincide Each geometric complex can be associated with an abstract complex by choosing as a ground set the set of vertices that appear in any simplex of and as system of subsets the subsets of which correspond to vertex sets of simplices in . A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: Consider for instance abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex: Let be an abstract simplicial complex above a set . Choose a union of simplices , but each in of dimension sufficiently large, such that the geometric simplex is of dimension if the abstract geometric simplex has dimension . If , can be identified with a face of and the resulting topological space is the gluing Effectuating the gluing for each inclusion, one ends up with the desired topological space. As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets being closed in the subspace topology of each simplex . The simplicial complex , which consists of all simplices of dimension is called the -th skeleton of . A natural neighborhood of a vertex of a simplicial complex is considered to be the star of a simplex, its boundary is the link . Simplicial maps The maps considered in this category are simplicial maps: Let , be abstract simplicial complexes above sets , . A simplicial map is a function which maps each simplex in onto a simplex in . By affine-linear extension on the simplices, induces a map between the geometric realizations of the complexes. Examples Let and let . The associated geometric complex is a star with center . Let and let . Its geometric realization is a tetrahedron. Let as above and let . The geometric simplicial complex is the boundary of a tetrahedron . Definition A triangulation of a topological space is a homeomorphism where is a simplicial complex. Topological spaces do not necessarily admit a triangulation and if they do, it is not necessarily unique. Examples Simplicial complexes can be triangulated by identity. Let be as in the examples seen above. The closed unit ball is homeomorphic to a tetrahedron so