446 A. Ultrafilters and UltraproductsExample A.3. Let I be an infinite set. The Frechet filter on I is the family
of all subsets whose complements are finite.
Example A.4. Let I be a directed set. The cofinal filter base on I is the
family of all subsets of the forms {i EI: i ::'.::: io} for some io EI.
Theorem A.5. Let I be a set and U' be a filter base on I. Then, there
exists an ultrafilter U which contains U'.Proof. Let U' be a filter base on I. By Zorn's Lemma, there exists a
maximal filter base U on I which contains U'. By maximality, U is a filter.
Now, let Io c I be given. We claim that if Io tj. U, then Ion J = 0 for some
J E U. Otherwise, the family U U {Io n J : J E U} would be a filter base
and would contradict the maximality of U. Since Jc I\ Io and J EU, this
means that Io tj. U implies I\ Io EU. D
Definition A.6. A cofinal ultrafilter U on a directed set I is an ultrafilter
which contains the cofinal filter base. It is also called a free or nonprincipal
ultrafilter when I= N, the directed set of natural numbers.The notion of filter comes from topology. It can be used to axiomatize
topological spaces, but it lost out to the open set formalism. Still it has some
advantages since it is easier to incorporate the Axiom of Choice ( ultrafilter)
into it.
Definition A. 7. Let X be a topological space. A net (xi)iEJ in X is said
to converge along the filter U on I if there exists x E X such that for any
open neighborhood 0 of x, the set {i EI: Xi E O} belongs to U. The limit
point x is denoted by limi-+U Xi· Observe that the limit point x is unique
provided that the topology is Hausdorff.
If U is a principal ultrafilter generated by io, then limi-+U Xi = Xio. If
I is a directed set and U is a cofinal ultrafilter, then the limit point (if it
exists) is a cluster point in the ordinary sense of topology. The following is
the first application of ultrafilters. We leave the proof as an exercise.
Theorem A.8. Let X be a Hausdorff topological space. The following are
equivalent:
(1) X is compact;
(2) any net in X converges along any ultrafilter on the index set;
(3) any net in X converges along some cofinal ultrafilter on the index
set.
Let I be a set and Ube an ultrafilter on I. Since bounded subsets of C
are pre-compact, the above theorem allows us to define a map <p: £^00 (I) ---+ C