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.