The Classification of the Filters and the Construction of the Free Ultrafilters

The classification of filter is discussed by analysis the relationship between the filter and the intersection of all members. It is obtained that on the infinite set there exists three types of filter, which are free, principal and preprincipal filter. It is proved that the sufficient and necessary condition of u is a free ultra filter by the properties of free filter. According to the above theorem, the special form of the free filter is constructed on Integers and Natures Set. Finally, the cardinal of the family of ulrta filters class is showed.