Abstract
Let E be a set with n elements, and let T(n,k) be the number of all labeled topologies having k open sets that can be defined on E. In this paper, we compute these numbers for k <= 17, and arbitrary n, as well as t(N0)(n,k), the number of all unlabeled non-T-0 topologies on E with k open sets, for 3 <= k <= 8.