An ultrafilter on a nonempty set 
is a nonempty collection 
of subsets of 
having the following properties:
1. 
.
2. If 
then 
.
3. If 
and 
then 
.
4. For any subset 
of 
,
either 
or its complement 
.
An ultrafilter 
on 
is said to be free if it contains the
cofinite filter 
of 
.
