Abstract

• In this paper, firstly the classification of filters is discussed. Topological space plays a crucial role in this discussion. After that the relations between them are studied. Especially, the characterizations of ultrafilters are presented with detail proofs. Ultrafilter is a powerful tool both in set theory and in topology. Moreover, the comparisons of filters are expressed and some notions of filter basis and trace of filter are described. Finally, ultrafilter convergence theorem and convergence of Cauchy filter in topological vector space are investigated.