a hypergraph that has the union of the sets as its vertices. An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face. In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family of finite sets without repetitions in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex. An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called the incidence relation, specifying which points belong to which lines. An incidence structure can be specified by a family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to each line, and any family of sets can be interpreted as an incidence structure in this way. A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length. When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A block code can also be described as a family of sets, by describing each codeword as the set of positions at which it contains a 1. A topological space consists of a pair where is a set (whose elements are called points) and is a on which is a family of sets (whose elements are called open sets) over that contains both the empty set and itself, and is closed under arbitrary set unions and finite set intersections. Covers and topologies A family of sets is said to a set if every point of belongs to some member of the family. A subfamily of a cover that continues to cover is called a . A family is called a if every point of lies in only finitely many family members. If every point lies in exactly one member then the cover is called a . When is a topological space, then a cover whose members are all open sets is called and . A family is called if each point in the space has a neighborhood that intersects only finitely many family members. A or is any family that is equal to a union of countably many