On the number of clopen topological spaces on a finite set

Let J(CL)(n) be the number of clopen topologies on a finite set of n elements. It is proved that the explicit formula for finding the total number of clopen topologies is E-n S-k=1(n) S (n, k) = i=1 CL(n, 2(i )), where S(n, k) is the Stirling number of the second kind and CL(n, 2i) is the number of clopen topologies having 2(i) open sets, i = 1, 2, 3, ..., n. Some results concerning the number of clopen topological spaces whose topologies have the same cardinality are also obtained.