Cofinite filters and directed set filters
We assume that is an infinite set.
The cofinite filter over is the set of subsets of such that is finite.
Since is infinite, is not in .
It is trivial to check that any superset in of an element of is in , and that the intersection of any two elements of is in . Thus is a filter over .
The cofinite filter over is defined independently of any ordering on .
If is an ultrafilter over , then either it is a principal ultrafilter, or finer than the cofinite filter, but not both.
Directed set filters
Directed set = ensemble ordonné filtrant, a partially ordered set such that for any , there exists with and .
The directed set filter is defined by the filter base ; that is that the elements of are the “segments” of all elements of greater than any .
If has no maximal element (that is, for all elements of , there is another element of that is strictly greater), then the directed set filter is finer than the cofinite filter.
Let be an element of the cofinite filter over .
Then is finite. Since is a directed set, there exists an that is greater than all elements of . There exists an that is strictly greater than all elements of . All elements in the segment are strictly greater, hence different from, all elements of , that is, that they belong to . Thus , and since belongs to the filter base , belongs to the filter .
Thus the cofinite filter is a subset of the directed set filter.