complete ultrafilter and partitions
If is an ultrafilter on a set , then
is -complete there is no partition
of into -many pieces for which each piece of the partition is not in .
We prove the case of -completeness; the case of arbitrary infinite cardinality follows closely. For the direction, let be a partition of into many pieces, all of which do not belong to , and write to illustrate this partition. Now, . Since, by our assumption
, each of the do not belong to , we have for each as is an ultrafilter. Thus, by -completeness. This, however, means , contradicting the definition of a filter.
Note that the converse states that every partition of into -many pieces has a (unique) piece . To prove this, let be a collection
of many members of and let . Now consider the partition of :
for each , put if is the least index for which .
It is easy to verify that each belongs to a unique , the collection of ’s is indeed a partition of .
Along with , partitions into many pieces. A (unique) piece of this partition belongs in : or . But, by the definition of . This excludes the possibility for the former to belong in (cf. alternative characterization of filter) and so .
Thus, starting from an arbitrary collection of -many members of , we have identified a partition of for which the unique piece which belongs to is . Therefore, is -complete.