locally finite families. One cover is said to another (coarser) cover if every member of is contained in some member of A is a particular type of refinement. Special types of set families A Sperner family is a set family in which none of the sets contains any of the others. Sperner's theorem bounds the maximum size of a Sperner family. A Helly family is a set family such that any minimal subfamily with empty intersection has bounded size. Helly's theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families. An abstract simplicial complex is a set family (consisting of finite sets) that is downward closed; that is, every subset of a set in is also in A matroid is an abstract simplicial complex with an additional property called the augmentation property. Every filter is a family of sets. A convexity space is a set family closed under arbitrary intersections and unions of chains (with respect to the inclusion relation). Other examples of set families are independence systems, greedoids, antimatroids, and bornological spaces. See also (or Set of sets that do not contain themselves) Notes References External links Basic concepts in set theory