inclusion, is always a Boolean algebra; hence the results of the above section apply. An (ultra)filter on is often called just an "(ultra)filter on ". The above formal definitions can be particularized to the powerset case as follows: Given an arbitrary set an ultrafilter on is a set consisting of subsets of such that: The empty set is not an element of If and are subsets of the set is a subset of and is an element of then is also an element of If and are elements of then so is the intersection of and If is a subset of then either or its relative complement is an element of Another way of looking at ultrafilters on a power set is as follows: for a given ultrafilter define a function on by setting if is an element of and otherwise. Such a function is called a 2-valued morphism. Then is finitely additive, and hence a on and every property of elements of is either true almost everywhere or false almost everywhere. However, is usually not , and hence does not define a measure in the usual sense. For a filter that is not an ultrafilter, one would say if and if leaving undefined elsewhere. Applications Ultrafilters on power sets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in Stone's representation theorem. In set theory ultrafilters are used to show that the axiom of constructibility is incompatible with the existence of a measurable cardinal . This is proved by taking the ultrapower of the set theoretical universe modulo a -complete, non-principal ultrafilter. The set of all ultrafilters of a poset can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element of , let This is most useful when is again a Boolean algebra, since in this situation the set of all is a base for a compact Hausdorff topology on . Especially, when considering the ultrafilters on a powerset the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality The ultraproduct construction in model theory uses ultrafilters to produce a new model starting from a sequence of -indexed models; for