## Cofinite filters and directed set filters

We assume that is an infinite set.

## Cofinite filters

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 .

### Property:

If is an ultrafilter over , then either it is a principal ultrafilter, or finer than the cofinite filter, but not both.

Proof:

## 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 .

### Property:

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.

Proof:

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.