To say a family of subsets of a set is independent means that there are no trivial Boolean combinations that can be formed from them except tautologically trivial ones.
Let be a set. A family (or ) is independent if, for any finite subfamily of pairwise distinct elements of , say , the intersection
is inhabited.
Another way of saying it: let denote the free Boolean algebra generated by the set . An inclusion names an independent family if the induced Boolean algebra map is itself injective. (To see how the alternative condition follows, consider elements of as written in disjoint normal form, i.e., as disjoint unions of finite intersections of literals ; use this to deduce ) This is likely the origin of the term “independent family”; cf. the abstract definition of linearly independent set.
Let be an infinite set, of cardinality .
The number of ultrafilters on is .
Begin by noting that the set of ultrafilters is a subset of , a set of cardinality . So it suffices to prove has at least elements.
The free Boolean algebra generated by also has cardinality . (Consider it as the directed union of ranging over finite subsets of , and use the fact that is finite.) The Stone space of , i.e., the spectrum? of consisting of Boolean ring homomorphisms , is the compactum . Let be the associated canonical pairing.
For each let . We claim that the form an independent family, i.e., for pairwise distinct , the finite intersection
is inhabited; this is equivalent to the claim that for pairwise distinct there exists such that and . To prove this, note that is an isomorphism by Stone duality, where is isomorphic to the Boolean algebra of clopens, and there exists a clopen separating the set of points from the set of points (a proof may be based on the lemma here, taking to be the complement of ).
By transport along a bijection , this result implies there is an independent family of subsets of that has size , or in other words there is an injective Boolean algebra map sending to . We finish by observing that given a Boolean algebra injection , any Boolean algebra map extends to a Boolean algebra map ; this follows from the ultrafilter principle as the ultrafilter in generates a proper filter in by injectivity of , which may then be extended to an ultrafilter in . (See also here: is an injective object.) From this observation, the induced map
is surjective, and thus the set of ultrafilters on has cardinality at least .
Last revised on March 18, 2016 at 17:10:16. See the history of this page for a list of all contributions to it.