On The Number Of Topologies On A Finite Set

In this paper we study some properties of T（n） and T₀（n）, the number of topologies and T₀ topologies respectively on an n element set.Firstly, we prove the period of T（n） modular 2, i.e., the parity of T（n）. Using correspondence between σ-algebras and partitions on a finite set, we getWe define a kind of topology and use C（n） to denote number of such topologies on an n element set. We get a recursive formula involving T（n）, C（n） and B（n）（number of σ-algebras on a finite set）,In section 1 we review results about topologies on a finite set. Then we give some basic definitions in section 2. We prove the two results above in section 3. We get some calculation results about T（n） and T₀（n） modulo powers of certain primes and put them in appendix.